Incarnations and Expansions of the Lord

Sri Krsna: the avatari
Dasavataras
Purusa-avataras
Lila-avataras
Guna-avataras
Manvantara-avataras
Yuga-avataras
Saktyavesa-avataras
Recognizing the real avatara

Sri Krsna: the avatari, the source of all avataras

ete cAMza-kalAH puMsaH kRSNas tu bhagavAn svayam
indrAri-vyAkulaM lokaM mRDayanti yuge yuge

All of the above-mentioned incarnations are either plenary portions or portions of the plenary portions of the Lord, but Lord Sri Krsna is the original Personality of Godhead. All of them appear on planets whenever there is a disturbance created by the atheists. The Lord incarnates to protect the theists. (SB 1.3.28)

According to Jiva Gosvami this is the main SB verse proving the superiority of Sri Krsna over various forms of Visnu. There are many other verses stating the same. See the sample below:

svayaM tv asAmyAtizayas tryadhIzaH svArAjya-lakSmy-Apta-samasta-kAmaH
baliM haradbhiz cira-loka-pAlaiH kirITa-koTyeDita-pAda-pIThaH

Lord Sri Krsna is the Lord of all kinds of threes and is independently supreme by achievement of all kinds of fortune. He is worshiped by the eternal maintainers of the creation, who offer Him the paraphernalia of worship by touching their millions of helmets to His feet. (SB 3.2.21), explanation in Krsna sandarbha Anuccheda 76

aSTamas tu tayor AsIt svayam eva hariH kila
subhadrA ca mahAbhAgA tava rAjan pitAmahI

The eighth son was the Supreme Personality of Godhead Himself—Krsna. The highly fortunate Subhadra, the one daughter, was your grandmother. (SB 9.24.55)

vasudeva-gRhe sAkSAd bhagavAn puruSaH paraH
janiSyate tat-priyArthaM sambhavantu sura-striyaH

The Supreme Personality of Godhead, Sri Krsna, who has full potency, will personally appear as the son of Vasudeva. Therefore all the wives of the demigods should also appear in order to satisfy Him. (SB 10.1.23)

matsyAzva-kacchapa-nRsiMha-varAha-haMsa-
rAjanya-vipra-vibudheSu kRtAvatAraH
tvaM pAsi nas tri-bhuvanaM ca yathAdhuneza
bhAraM bhuvo hara yadUttama vandanaM te

O supreme controller, Your Lordship previously accepted incarnations as a fish, a horse, a tortoise, Narasimhadeva, a boar, a swan, Lord Ramacandra, Parasurama and, among the demigods, Vamanadeva, to protect the entire world by Your mercy. Now please protect us again by Your mercy by diminishing the disturbances in this world. O Krishna, best of the Yadus, we respectfully offer our obeisances unto You. (SB 10.2.40)

bahUni santi nAmAni rUpANi ca sutasya te
guNa-karmAnurUpANi tAny ahaM veda no janAH

For this son of yours there are many forms and names according to His transcendental qualities and activities. These are known to me, but people in general do not understand them. (SB 10.8.15)

yasyAvatArA jJAyante zarIreSv azarIriNaH
tais tair atulyAtizayair vIryair dehiSv asaGgataiH

Appearing in bodies like those of an ordinary fish, tortoise and hog, You exhibit activities impossible for such creatures to perform - extraordinary, incomparable, transcendental activities of unlimited power and strength. These bodies of Yours, therefore, are not made of material elements, but are incarnations of Your Supreme Personality. You are the same Supreme Personality of Godhead, who have now appeared, with full potency, for the benefit of all living entities within this material world. (SB 10.10.34)

nArAyaNas tvaM na hi sarva-dehinAm
AtmAsy adhIzAkhila-loka-sAkSI
nArAyaNo ’GgaM nara-bhH-jalAyanAt
tac cApi satyaH na tavaiva mAyA

Are You not the original NArAyaNa, O supreme controller, since You are the soul of every embodied being and the eternal witness of all created realms? Indeed, Lord NArAyaNa is Your expansion, and He is called NArAyaNa because He is the generating source of the primeval water of the universe. He is real, not a product of Your illusory Maya. (SB 10.14.14)

jAnatAM tvat-padavIm anAtmany
AtmAtmanA bhAsi vitatya mAyAm
sRSTAv ivAhaM jagato vidhAna
iva tvam eSo 'nta iva trinetraH

To persons ignorant of Your actual transcendental position, You appear as part of the material world, manifesting Yourself by the expansion of Your inconceivable energy. Thus for the creation of the universe You appear as me [Brahma], for its maintenance You appear as Yourself [Vishnu], and for its annihilation You appear as Lord Trinetra [Siva]. (SB 10.14.19)

sureSv RSiSv Iza tathaiva nRSv api
tiryakSu yAdaHsv api te 'janasya
janmAsatAM durmada-nigrahAya
prabho vidhAtaH sad-anugrahAya ca

O Lord, O supreme creator and master, You have no material birth, yet to defeat the false pride of the faithless demons and show mercy to Your saintly devotees, You take birth among the demigods, sages, human beings, animals and even the aquatics. (SB 10.14.20)

aho bhAgyam aho bhAgyaM nanda-gopa-vrajaukasAm
yan-mitraM paramAnandaM pUrNaM brahma sanAtanam

How greatly fortunate are Nanda Maharaja, the cowherd men and all the other inhabitants of Vrajabhumi! There is no limit to their good fortune, because the Absolute Truth, the source of transcendental bliss, the eternal Supreme Brahman, has become their friend. (SB 10.14.32)

nAyaM zriyo 'Gga u nitAnta-rateH prasAdaH
svar-yoSitAM nalina-gandha-rucAM kuto 'nyAH
rAsotsave 'sya bhuja-daNDa-gRhIta-kaNTha-
labdhAziSAM ya udagAd vraja-sundarINAm

The gopis received benedictions from the Lord that neither Laksmidevi nor the most beautiful dancers in the heavenly planets could attain. In the rasa dance, the Lord showed His favor to the most fortunate gopis by placing His arms on their shoulders and dancing with each of them individually. No one can compare with the gopis, who received the causeless mercy of the Lord. (SB 10.47.60)

Note: This verse says Laksmi could not join Krsna in rasa lila although SB 2.4.20 describes Sri Krsna as sriyah pati, husband of Sri, Laksmi. This is to show the difference in rasas, madhurya being superior to vatsalya.

yat-pAda-paGkaja-rajaH zirasA bibharti
zRIr abja-jaH sa-girizaH saha loka-pAlaiH
lIlA-tanuH sva-kRta-setu-parIpsayA yaH
kAle 'dadhat sa bhagavAn mama kena tuSyet

Goddess Laksmi, Lord Brahma, Lord Siva and the rulers of the various planets place the dust of His lotus feet on their heads, and to protect the codes of religion, which He has created, He assumes pastime incarnations at various times. How may that Supreme Personality of Godhead become pleased with me? (SB 10.58.37)

yasyAMzAMzAMza-bhAgena vizvotpatti-layodayAH
bhavanti kila vizvAtmaMs taM tvAdyAhaM gatiM gatA

O Soul of all that be, the creation, maintenance and destruction of the universe are all carried out by a fraction of an expansion of an expansion of Your expansion. Today I have come to take shelter of You, the Supreme Lord. (SB 10.85.31)

zrI-nArada uvAca
namas tasmai bhagavate kRSNAyAmala-kIrtaye
yo dhatte sarva-bhUtAnAm abhavAyozatIH kalAH

Sri Narada said: I offer my obeisances to Him of spotless fame, the supreme Lord Krsna, who manifests His all-attractive personal expansions so that all living beings can achieve liberation. (SB 10.87.46)

ojaH saho balavatAM karmAhaM viddhi sAtvatAm
sAtvatAM nava-mUrtInAm Adi-mUrtir ahaM parA

Of the powerful I am bodily and mental strength, and I am the devotional activities of My devotees. My devotees worship Me in nine different forms, among which I am the original and primary Vasudeva. (SB 11.16.32)

zrI-zuka uvAca
ity uddhavenAty-anurakta-cetasA
pRSTo jagat-krIDanakaH sva-zaktibhiH
gRhIta-mUrti-traya Izvarezvaro
jagAda sa-prema-manohara-smitaH

Sukadeva Gosvami said: Thus questioned by the most affectionate Uddhava, Lord Krishna, the supreme controller of all controllers, who takes the entire universe as His plaything and assumes the three forms of Brahma, Visnu and Siva, began to reply, lovingly displaying His all-attractive smile. (SB 11.29.7)

bhava-bhayam apahantuM jJAna-vijJAna-sAraM
nigama-kRd upajahre bhRGga-vad veda-sAram
amRtam udadhitaz cApAyayad bhRtya-vargAn
puruSam RSabham AdyaM kRSNa-saMjJaM nato 'smi

I offer my obeisances to that Supreme Personality of Godhead, the original and greatest of all beings, Lord Sri Krsna. He is the author of the Vedas, and just to destroy His devotees' fear of material existence, like a bee He has collected this nectarean essence of all knowledge and self-realization. Thus He has awarded to His many devotees this nectar from the ocean of bliss, and by His mercy they have drunk it. (SB 11.29.49)

na brahmaNo bhidA vipra zrI kRSNasya ca sattama
nArAyaNasya vA saumya hy avatAri svarUpiNaH

O brahmana, O best of the devotees, these divisions do not apply to the impersonal Brahman. Neither do they apply to Lord Krsna, who is the original Supreme Personality of Godhead and the source from which the many incarnations have come. (Satvata tantra 3.35)

ekam eva paraM tattvaM avatAri-sanAtanam
zrI-kRSNa-brahma-puruSaiH samjJAbhir gIyate pRthak

The eternal source of the incarnations is one, although He is variously called Sri Krsna, Brahman, and the purusa incarnation or Supersoul. (Satvata tantra 3.39)

ataH sAtvata-tantra-jJA bhakti-niSThA vilakSaNAH
zrI-kRSNAkhyaM paraM dhAma paramAnandam uttamam

The great souls that understand the Satvata Tantra and have faith in devotional service say that supremely blissful and glorious Sri Krsna is the highest. (Satvata tantra 3.41)

ataH sarvAvatArANAM kAraNaM kRSNa ucyate
sRSTy-Ady-aneka-karmANi darzitAni yataH svataH

Therefore Lord Krsna, who displays many different activities beginning with the creation of the material worlds, is said to be the origin of all incarnations. (Satvata tantra 3.52)

rAmAdi-mUrtiSu kalA-niyamena tiSThan
nänävatAram akarod bhuvaneSu kintu
kRSNaH svayaM samabhavat paramaH pumAn yo
govindam Adi-puruSaM tam ahaM bhajAmi

I worship the Supreme Personality of Godhead, Govinda, who is always situated in various incarnations such as Rama, Nrsimha and many sub-incarnations as well but who is the original Personality of Godhead known as Krsna and who incarnates personally also. (Brahma samhita 5.39)

yasyaika-nizvasita-kAlam athAvalambya
jIvanti loma-vila-jA jagad-aNDa-nAthAH
viSNur mahAn sa iha yasya kalA-vizeSo
govindam Adi-puruSaM tam ahaM bhajAmi

The Maha-Visnu, into whom all the innumerable universes enter and from whom they come forth again simply by his breathing process, is a plenary expansion of Krsna. therefore i worship Govinda, Krsna, the cause of all causes. (Brahma samhita 5.48)

devi sarve 'vatArAs tu brahmanaH kRSNa rUpiNaH
avatArI svayaM kRSNaH saguNo nirguNaH svayam

O goddesss, all avataras come from Brahman in the form of Krsna Himself, the avatari. He is both with and without gunas. (Narada Purana 2.58.45)

zrI-varuNa uvAca
namaH zrI-kRSNacandrAya paripUrNatamAya ca
asaGkhya-brahmANDa-bhRte goloka-pataye namaH

Sri Varuna said: Obeisances to Sri Krsnacandra, the perfect Supreme Personality of Godhead! Obeisances to the Lord of Goloka, the maintainer of numberless universes! (Garga samhita 4.22.5)

yatrAnena sva-bhrAtrApi vArtA syAd 'dhi parasparam
iti pravadatas tAn vai nitvA zrI-bhagavAn hariH
vrajam AgatavAn rAjan paripUrNatamaH prabhuH

O king, then they said among themselves, "The Supreme Personality of Godhead and His brother have come to Vraja!" (Garga samhita 4.22.19)

zrI-nArada uvAca
paripUrNatamaH sAkSAd bhagavAn bhakta-vatsalaH
sasmara gokulam dInaM goSa-gopAla-saGkulaM

Sri Narada said: Then the original Supreme Personality of Godhead, who loves His devotees, remembered unhappy Gokula, filled with cows and cowherd people. (Garga samhita 5.13.2)

paripUrNatame sAksAc chrI-kRSNe puruSottame
IdrcI ca krtA bhaktir yuvabhyAm prema-lakSaNa
tIrthAtana-tapo-dAna- saGkhyA-yogaiz ca durlabha
zAzvati yuvayoH prApta yA bhaktiH prema-lakSaNa

The eternal love and devotion you both feel for Lord Krsna, the Supreme Personality of Godhead, cannot be attained by pilgrimages, austerity, charity, sagkhya, or yoga. (Garga samhita 5.14.33-34)

paripUrNatamaM viddhi zrI-kRSNam nanda-nandanam
kamsAdInaM vadhArthAya bhaktAnAm rakSanAya ca
brAhmaNa prArthitah kRSNo 'vatatAra gRhe tava
jAta-mAtro 'dbhutam lIlam cakAra sa-balo hariH

Know that Nanda's son, Krsna, is the original Supreme Personality of Godhead. Requested by the demigod Brahma, and in order to protect the devotees and kill the demons led by Kamsa, He descended to your home. Since the moment of His birth, He and Balarama enjoyed many wonderful pastimes. (Garga samhita 5.14.37-38)

zrI-uddhava uvAca
sadAsti kRSNaH paripUrNa-devo rAdhe sadA tvaM paripUrNa-devI
zrI-kRSNacandraH kRta-nitya-lIlo lIlAvatI tvaM kRta-nitya-lIlA

Sri Uddhava said: Eternally Sri Krsna is the perfect Supreme Personality of Godhead. Eternally You are His supreme goddess. Eternally Sri Krsna enjoys transcendental pastimes. Eternally You enjoy transcendental pastimes with Him. (Garga samhita 5.15.22)

zrI-devaGgana Ucuh
aMzAMzakamzaka-kalAdy-avatAra-vRndair
Aveza-pUrNa-sahitaz ca parasya yasya
sargadayah kila bhavanti tam eva kRSNam
pUrNAt paraM tu paripUrNatamaM nataH smaH

The gopis that had been demigoddesses said: The amsa, amsamsa, kala, avesa, purna, and other incarnations of Godhead come from Him. The creation, maintenance, and destruction of the material universes come from Him. We bow down before Him, Sri Krsna, the original Supreme Personality of Godhead, more perfect than the perfect. (Garga samhita 5.17.23)

(There are many more such paripUrNatama quotes in Garga samhita. In Garga samhita 8.13.11 is also Balarama called "paripUrNatamaH sAkSAt paramaH puruSottamaH" which is in sync with SB 1.1.18-20 so the plural mRdayanti in SB 1.3.28 makes sense.)

The gradation in the power of holy names is sometimes doubted but it is supported by sastras:

mahAdeva uvAca
zrI rAma rAma rAmeti rAme rAme manorame
sahasra nAma tat tulyaM rAma nAma varAnane

Lord Siva tells to his consort Devi Parvati: "O Delight of my heart, o beautiful-faced lady! I myself constantly enjoy the supreme bliss chanting the divine Name of my beloved as 'Sri Rama, Sri Rama' which is equal to reciting one thousand Names of Lord Visnu, Sriman Narayana." (Brhad Visnu Sahasra Nama stotra, Padma Purana 6.71.331 MUM e-edition p. 2548; 6.254.20 MUM e-edition p. 3335). More refs: Narada Pancaratra 4.3.223, quoted in: CC 2.9.32-33 p., Bhaktivinod Thakur - Harinama Cintamani 8; SB 1.19.6 p.

sahasra-nAmnAM puNyAnAM trir-AvRttyA tu yat phalaM
ekAvRttyA tu kRSNasya nAmaikaM tat prayacchati

The pious results (punya) achieved by chanting the thousand holy names of Visnu (Visnu-sahasra-nama-stotram) three times can be attained by only one utterance of the holy name of Krsna. (Brahmanda Purana Upodhgatapada 36.19)

kRSNa-caitanya-nAmnA ye kIrtayanti sakRn naraH
nAnAparAdha-muktAs te punanti sakalam jagat

Those who chant the name of Krsna Caitanya even once, become free from all offenses and purify all the worlds by their very presence. (Visnu Yamala)

Another point to consider: Visnu sahasra nama (from Mahabharata Anusasana Parva ch. 135) doesn't include any names of Krsna related to His rasa lila, yet various Krsna-related sahasranamas like Gopala sahasra nama from Sammohana tantra include them, together with the names of other forms of Krsna/Visnu. That alludes to the superior position of Vrindavan Krsna vs. all other forms.

Related:
The supreme position of Mathura/Vrndavan (even above Vaikuntha)
The standard of valid knowledge (based on Sri Jiva Goswami's Tattva-sandarbha, 9-26)

Lord Caitanya explained to Sanatana Gosvami that the expansions of Lord Krsna who come to the material creation are called avataras. The word 'avatara' means 'one who descends' (from Sanskrit avatarati) and in this case the word specifically refers to one who descends from the spiritual sky. In the spiritual sky there are innumerable Vaikuntha planets, and from these planets the expansions of the Supreme Personality of Godhead come into this universe.

Sometimes the word 'incarnation' (lit. 'one who entered flesh', ie. accepted a material body) is used but this word from the theological point of view specifically refers to Jesus (who is called "the Only Prophet Has Appeared in Different Ages", i.e. in essence an avatar, in Clementine Homilies 3, Chapter XX.: "But give heed to my first discourse of the truth. If any one do not allow the man fashioned by the hands of God to have had the Holy Spirit of Christ, how is he not guilty of the greatest impiety in allowing another born of an impure stock to have it? But he would act most piously, if he should not allow to another to have it, but should say that he alone has it, who has changed his forms and his names from the beginning of the world, and so reappeared again and again in the world, until coming upon his own times, and being anointed with mercy for the works of God, he shall enjoy rest forever. His honour it is to bear rule and lordship over all things, in air, earth, and waters. But in addition to these, himself having made man, he had breath, the indescribable garment of the soul, that he might be able to be immortal.") Srimad Bhagavatam 5.5.19 describes the body of the Lord as 'durvibhavya', materially inconceivable, since it may sometimes seem to be material (SB 10.14.55) or the Lord may manifest a material body for some purpose if it is required in the lila (like when He 'left His body' at the end of Dvaraka-lila). This type of body shouldn't be mistaken as a usual human body. Also, humans and other jivas cannot themselves manifest avataras (if they are endowed with specific potency, they are called saktyavesa-avataras though). Using this word for icons in internet forums, etc., is therefore incorrect. ;)

There are six kinds of incarnations: (1) the purusa-avataras, (2) the lila-avataras, (3) the guna-avataras, (4) the manvantara-avataras, (5) the yuga-avataras, and (6) the saktyavesa-avataras.

Lord Caitanya explains some of them to Sanatana Gosvami just to give him an idea of how the Lord expands and enjoys. These conclusions are also confirmed in Srimad-Bhagavatam (1.3.26). There it is said that there is no limit to the incarnations of the Supreme Lord, just as there is no limit to the waves of the ocean.

Another description of avataras is found in conversation between Narada Muni and king Bahulasva recorded in Garga samhita 1.1.16-27:

zrI-nArada uvAca
aMzAMzo 'Mzas tathAvezaH kalA pUrNaH prakathyate
vyAsAdyaiz ca smRTaH SaSThaH paripUrNatamaH svayam

Sri Narada said: In the smrti-sastra the great sages who have Vyasa as their leader explain that the Lord descends in six kinds of forms: 1. amsamsa (a part of a part), 2. amsa (a part), 3. avesa (entrance into a jiva), 4. kala (a full part), 5. purna (full), and 6. paripurnatama (most full). 1.1.16

aMzAMzas tu marIcy-Adir aMzA brahmAdayas tathA
kalAH kapila-kUrmAdyA AvezA bhArgavAdayaH

The amsamsa incarnations begin with Marici, the amsa incarnations begin with Brahma, the kala incarnations begin with Kapila and Kurma, and the avesa incarnations begin with Parasurama. 1.1.17

pUrNo nRsiMho rAmaz ca zvetadvIpAdhipo hariH
vaikuNTho 'pi tathA yajJo nara-nArAyaNaH smRtaH

The smrti-sastra explains that the purna incarnations are 1. Nrsimha, 2. Rama, 3. Lord Hari, the ruler of Svetadvipa, 4. Vaikuntha, 5. Yajna, and 6. Nara-Narayana. 1.1.18

paripUrNatamaH sAkSAc chrI-kRSNo bhagavAn svayam
asaGkhya-brahmANDa-patir goloke dhAmni rAjate

The paripurnatama form of the Lord is Sri Krsna, who is the original Supreme Personality of Godhead Himself. He is the master of countless universes. He is splendidly manifest in the realm of Goloka. 1.1.19

kAryAdhikAraM kurvantaH sad-aMzAz te prakirtitAH
tat-kArya-bhAraM kurvantas te 'MzAMzA viditAH prabhoH

The amsa incarnations are said to oversee the execution of the Lord's mission (in the world). The amsamsa incarnations are understood to perform the work of executing those missions. 1.1.20

yeSAm antar-gato viSNuH kAryaM kRtvA vinirgataH
nAnAveSAvatArAMz ca viddhi rAjan mahA-mate

O great soul, O king, please know that in the various avesa incarnations Lord Visnu enters (a jiva), executes His mission, and then departs. 1.1.21

dharmaM vijJAya kRtvA yaH punar antaradhIyata
yuge yuge vartamAnaH so 'vatAraH kalA hareH

Millennium after millennium Lord Hari's kala incarnations teach and establish the principles of religion, and then disappear. 1.1.22

catur-vyUho bhaved yatra dRzyante ca rasA nava
ataH paraM ca vIryANi sa tu pUrNaH prakathyate

The purna incarnations are said to include the catur-vyuha incarnations. They are said to be the places where heroic powers and the nine rasas are seen. 1.1.23

yasmin sarvANi tejAMsi vilIyante sva-tejasi
taM vadanti pare sAkSAt paripUrNatamaM svayam

Others say that the form of the Lord where all powers and glories enter is His paripurnatama form, the original form of the Lord Himself. 1.1.24

pUrNasya lakSaNaM yatra taM pazyanti pRthak pRthak
bhAvenApi janAH so 'yaM paripUrNatamaH svayam

The transcendental qualities people see distributed among the Lord's purna incarnations are all present in the Lord's original, paripurnatama form. 1.1.25

paripUrNatamaH sAkSAc chrI-kRSNo nAnya eva hi
eka-kAryArtham Agatya koTi-kAryaM cakAra ha

The Lord's original, paripurnatama form is Sri Krsna and no one else. Coming (to this world) to execute one mission, He executes millions of missions. 1.1.26

pUrNaH purANaH puruSottamottamaH
parAt paro yaH puruSaH parezvaraH
svayaM sadAnandamayaM kRpAkaraM
guNAkaraM taM zaraNaM vrajAmy aham

He is perfect and complete. He is the oldest. He is the most exalted of exalted persons. He is greater than the greatest. He is the Supreme Person. He is the master of the exalted. Of Him who is full of bliss, who is jewel-mine of mercy, who is a jewel-mine of transcendental virtues, I take shelter. 1.1.27

ye yadu nandana nihatAs te tu na bhUyaH punar bhavaM prApuH
tasmAd avatArANAm antaryAmI pravartakaH kRSNaH

As regards those who were killed by (Krsna), the delight of the Yadu-race, they did not again suffer rebirth. Therefore, Krsna is the inner controller (antaryami) and originator (pravartaka) of avataras. (Prabodha sudhakara 241 by Adi Sankaracarya)

Dasavataras

Q: Lord Buddha described as one of the Dasavataras and as a saktyavesa avatara. Which is correct? What kind of avatar is Lord Ramacandra?

A: Both are correct. The word "dasavatara" or "dasa-avatara" denotes not a category of avataras like yuga-, lila-, manvantara-, or other avataras but the most famous list of ten avataras (in Sanskrit "dasa" = "10"). This list is to be found e.g. in Garuda Purana 1.86.10-11 (or, according to other numbering, 2.30.37):
"The names of the ten incarnations of the Lord [dasavatara], viz - Matsya (fish), Kurma (tortoise), Varaha (boar), Narasimha, Srirama, Parasurama, Krsna, Balarama, Buddha and Kalki shall be remembered always."
There are other lists as well, like 25 in Bhagavata Purana 1.3 but these ten are the most popular ones. Narada Purana 1.62.52-54 includes most of them as well. They are the ones most often depicted by artists and celebrated by poets like Jayadeva Gosvami in his Dasavatara-stotra. Here is a movie for children about Dasavatara.

Saktyavesa-avataras is a category of avataras who are not visnu-tattva but jiva-tattva (jivas, or living beings, empowered for a special mission).

The system of avatara categories is quite complex and some avataras belong to more than one category. If you wish to delve into this technical topic, consult books like Caitanya-caritamrta by Krsnadasa Kaviraja Gosvami (published and distributed by ISKCON), Laghu-bhagavatamrta by Rupa Gosvami, etc.

Dasavataras are practically all lila-avataras.

Sri Rama is the famous lila-avatara, also listed among those 25 in Bhagavata Purana. His lila is described in the epic Ramayana.

Purusa-avataras and their expansions

Krsna first incarnates as the three purusa-avataras, namely the Maha-Visnu or Karanodakasayi avatara, the Garbhodakasayi avatara and the Ksirodakasayi avatara. This is confirmed in the Satvata-tantra [specified as Naradiya tantra in Bhavartha-dipika 2.6.38 by Sridhara Svami, see also SB 1.3.8]:

visnos tu trini rupani
purusakhyany atho viduh
ekam tu mahatah srastr
dvitiyam tv anda-samsthitam
trtiyam sarva-bhuta-stham
tani jnatva vimucyate

"For material creation, Lord Krsna's plenary expansion assumes three Visnus. The first one, Maha Visnu, creates the total material energy, known as mahat-tattva. The second, Garbhodakasayi Visnu, enters into all the universes to create diversities in each of them. The third, Ksirodakasayi Visnu, is diffused as the all-pervading Supersoul in all the universes and is known as Paramatma. He is present even within the atoms. Anyone who knows these three Visnus can be liberated from material entanglement." (Satvata-tantra quoted in BG 7.4 purport)

Further references for these purusas:

BG 11.15

Srimad Bhagavatam 1.3.2-3, 2.2.8, 2.2.25, 2.6.42, 3.8.10,15,23, 3.20.15-16, 4.9.14, 9.14.2, 10.1.21, 10.14.14, 11.4.3, 11.15.16 [Bhavartha-dipika: "The Lord has three forms: 1. Ksirodakasayi Visnu, 2. Garbhodakasayi Visnu, and Karanodakasayi Visnu. His fourth form is Lord Vasudeva. That is why Lord Vasudeva is called 'turiya' (the fourth)."].

"Observing that Lord Ksirodakasayi Visnu's crown had been stolen by the demons, Garuda attacked the thieves, killed them, and recovered the Lord's crown. As all this was happening, Lord Ksirodakasayi Visnu descended to the earth where Lord Krsna was performing His pastimes, and entered Lord Krsna's body. In this way Lord Ksirodakasayi Visnu appeared during Lord Krsna's incarnation. Garuda, trying the return the crown, was unable to find his master in the spiritual world. He finally found Ksirodakasayi Visnu manifested within the form of Lord Krsna (who was then enjoying pastimes on the peak of Gomanta Mountain). Garuda then returned his master's crown." (Hari-vamsa, Visnu-parva, chapter 41)

Brahmavaivarta Purana 4.67.55-58 (all three mentioned, along with their Laksmis, as mahat-virat/Mahavisnu, ksudra-virat/Garbhodakasayi Visnu and the one who resides in the pore of ksudra-virat/Ksirodakasayi Visnu, 57):

bhedah kadapi na bhaven
niscitam ca tathavayoh
aham mahan virat srstau
visvani yasya lomasu

amsas tvam tatra mahati
svamsena tasya kamini
aham ksudra-virat srstau
visvam yan-nabhi-padmatah

ayam visnor loma-kupe
vaso me camsatah sati
tasya stri tvam ca brhati
svamsena subhaga tatha

tasya visve ca pratyekam
brahma-visnu-sivadayah
brahma-visnu-siva amsas
canyas capi ca mat-kalah

Brahmavaivarta Purana 4.13.138 (visnuh ksirodam eva ca)

Krsna is the original Personality of Godhead. His expansion is Balarama, who expands the original catur vyuha or quadruple expansions.

1) Vasudeva 2) Sankarsana 3) Pradyumna 4) Aniruddha

These original catur vyuha expansions reside in Mathura and Dvaraka. From them twenty four forms of Visnu expand. They are named differently according to the arrangement of the conch, disc, lotus and club in Their hands. All of these twenty four forms reside in each Vaikuntha planet with the predominating Deity of that planet. From the original catur vyuha the second catur vyuha expands. In this second catur vyuha the form of Sankarsana is also called Maha Sankarsana. It is from Maha Sankarsana that Maha Visnu becomes manifested.

KRSNA
Original Personality of Godhead
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BALARAMA
First expansion
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ORIGINAL CATUR VYUHA
Vasudeva, Sankarsana, Pradyumna, Aniruddha
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SECOND CATUR VYUHA
Vasudeva, Sankarsana,¿ Pradyumna, Aniruddha
(each expand into three and³then another two expansions)
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(Maha Sankarsana)
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MAHA VISNU

1) MAHA VISNU lies on the Causal Ocean which appears in one corner of the spiritual world. He manifests the mahat tattva (or the sum total of material energy). It is into this mahat tattva that He exhales all of the seedlike universes through the pores of His skin. These seedlike universes then expand as the different material elements form coverings around them. Each of the coverings is ten times thicker than the previous covering and form a shell-like covering. When Maha Visnu impregnates the living entities by His glance into the material nature it begins to manifest its various energies.

2) When the universes have thus developed in the womb of material nature, Maha Visnu expands as GARBHODAKASAYI VISNU, Who enters into each universe with the living entities of that particular universe and thus activates each universe by His presence. He lies down on Ananta Sesa who lies on the Garbha ocean, which is the perspiration from the Lord's body and half fills the universe. From His navel comes a lotus bud which is the total form of the living entities' fruitive activity. The lotus grows dissipating the darkness of the universe. On top of the lotus Brahma, the first living being appears. Situated on the lotus, Brahma could not understand anything. He began entering the stem and climbed down to find its origin. Not finding anything, Brahma again returned to the top of the lotus where he heard the word ta-pa (austerity). Hearing the sound Brahma underwent penance for one thousand celestial years; (6x30x12x1000 earthly years). Being very pleased with Brahma's tapasya, the Lord manifested the Vaikuntha planets to him. Seeing Vaikuntha, Brahma became very happy and bowed to the Lord. Being very pleased with Brahma the Lord shakes his hand and reveals how to create the universe.

3) To maintain the universe Garbhodakasayi Visnu expands as KSIRODAKASAYI VISNU, Who is the all-pervading Supersoul (Paramatma). By His entering into every atom He maintains the whole universe. His abode is Svetadvipa, an island in the ocean of milk.

Krsna's energies can also be divided into three: His energy of thinking feeling and acting. When He exhibits His thinking energy, He is the Supreme Lord; when He exhibits His feeling energy, He is Lord Vasudeva; when He exhibits His acting energy, He is Sankarsana Balarama. Without His thinking, feeling and acting, there would be no possibility of creation. Although there is no creation in the spiritual world - for there the planets are beginningless - there is creation in the material world. In either case, however, both the spiritual and material worlds are manifestations of the energy of acting, in which Krsna acts in the form of Sankarsana and Balarama.

References: SB 1.3.1-5, 2.9.4-9, 3.8.11-21, 10.46.31
Expansions and avataras chart (.jpg picture)

Lila-avataras (pastime avataras)

Because almost all of these twenty-five lila-avataras [listed in SB 1.3] appear in one day of Brahma, which is called a kalpa, they are sometimes called kalpa-avataras. Out of these, the incarnation of Hamsa and Mohini are not permanent, but Kapila, Dattatreya, Rsabha, Dhanvantari and Vyasa are five eternal forms, and they are more celebrated. Kurma (the tortoise incarnation), Matsya (the fish), Nara-narayana, Varaha (the boar), Hayasirsa, Prsnigarbha, and Balarama are considered incarnations of vaibhava. (TLC 7)

"Not permanent" means naimittika, "eternal" means nitya:

So far the avataras are concerned there are two types. One is called nitya and the other is called naimittic [naimittika]. Nitya means eternal and naimittic avataras appear for some specific function in the material worlds. Nitya-avataras appear for some specific function in the material worlds. Nitya-avatars have their eternal abodes in the spiritual sky from which they may sometimes descend to the material worlds, but naimittic avataras are expansions of nitya-avataras for some timely purpose. So the non-human forms of avataras do not have their planets in the spiritual sky.” (SPL to Ekayani, 25th July, 1970)

1) Catuhsana - The four Kumaras, sons of Lord Brahma

They appeared in the beginning of creation and are specifically empowered to distribute transcendental knowledge. They are empowered with the Lord's jnana-sakti and therefore they are also known as saktyavesa-avataras.

2) Narada Muni - The son of Lord Brahma

He is empowered with the Lord's bhakti-sakti and therefore he is also saktyavesa-avatara.

3) Varahadeva - Sukara or the Boar incarnation

The Boar incarnation appeared in two different millenniums. During the period of Svayambhuva Manu, the earthly planet remained submerged in the water of devastation, so the Lord appeared as a white boar and lifted the earth and set it properly. During the period of Caksusa Manu Lord Boar was red and He killed the demon Hiranyaksa.

4) Matsya - The Fish incarnation

He appeared to show special mercy to Satyavrata Muni. After the period of Caksusa Manu when there was a partial inundation, he also protected Satyavrata Muni (who later became Vaivasvata Manu) by keeping him safe on a boat.

5) Yajna - The son of Prajapati Ruci and Akuti

During the period of Svayambhuva Manu there was no qualified Indra. So the Lord appeared as Yajna and took that position.

6) Nara Narayana - The twin sons of King Dharma and Murti

They are partial expansions of Krsna and Arjuna. Nara and Narayana Rsis exhibited the Lord's renunciation.

7) Kapiladeva - The son of Kardama Muni and Devahuti

He explained the original (personal) Sankhya philosophy. Kapiladeva exhibited the Lord's transcendental knowledge.

8) Dattatreya - The son of Atri Muni and Anasuya

He is a combined incarnation of Lord Visnu, Brahma and Siva. He spoke on the subject of transcendence to Alarka, Prahlada, Yadu, etc.

9) Hayasirsa - Lord Hayagriva

At the end of the millennium ignorance personified took the form of a demon, stole the Vedas and took them to the planet Rasatala. At that time the Lord, at the request of Brahma, retrieved them after assuming the form of a horse.

10) Hamsa - The Swan incarnation

Due to bewilderment, Brahma, could not answer some questions put forward by his sons. At that time the Lord appeared as a swan and answered the questions.

11) Prsnigarbha - The incarnation who appeared before Dhruva

He created the planet known as Dhruvaloka for the habitation of Dhruva Maharaja.

12) Rsabha - The son of King Nabhi and Merudevi

His oldest son was Bharata Maharaja of which the earth was named after as Bharat-varsa. He instructed his sons to follow the path of perfection by tapasya (asceticism).

13) Prthu - The incarnation of the Lord's ruling force

Due to the demonic nature of King Vena, the sages killed him by a curse. They prayed for the Lord to appear and churned the two arms of the dead body according to a specific method and the Lord appeared as Prthu.

14) Nrsimhadeva - The half-man half-lion incarnation

He appeared to protect Prahlada by killing his demoniac father Hiranyakasipu. Sri Nrsimha manifests various bhavas, mostly wrathful but also peaceful ones (like some of His murtis in Ahobilam or Santa Nrsimha in Puri). Gaudiya Vaisnava siddhanta about Sri Nrsimha appearing in this universe is that He's not a form of Sri Visnu residing in one of Vaikunthas but directly Sri Krsna. His lila manifests only in the material realm, although perpetually and in that sense is also eternal (Prahlada Maharaja worships Krsna, not Narayana. Nrsimha Purana 41.36 says that on his slate he has written the name 'Krsna' and he remembers Krsna nama.). This is hinted on in SB 3.24.33 p., last par., and BG 4.8. He personifies this BG verse.

nRsiMha-rAma-kRSNesu sad-guNyam paripUritam: "In Narasimha, Rama and Krsna, all the six opulences (strength, wealth, renunciation, splendor, energy, wisdom) are fully manifest." (Padma Purana, quoted by Srila Rupa Gosvami in Laghu-bhagavatamrta 1.5.16)

Glories and worship of Nrsimha are described in Narada Purana 2.55.73-132 and Brahma Purana 58.4-70.

Srila Bhaktivinoda Thakura has written five beautiful prayers in "Sri Navadvipa Bhava Taranga" for receiving the mercy of Lord Nrsimha. These prayers are certainly assurance to all sincere devotees that the worship of Lord Nrsimha is purely in the line of aspiring love and devotion to Sri Sri Radha and Krsna.

e dusta hrdaye kama adi ripu chaya
kutinati pratisthasa sathya sada raya
hrdaya-sodhana ara krsnera vasana
nrsimha-carane mora ei to' kamana
Within my sinful heart the six enemies headed by lust perpetually reside, as well as duplicity, the desire for fame, plus sheer cunning. At the lotus feet of Lord Narasimha, I hope that He will mercifully purify my heart and give me the desire to serve Lord Krsna.
kandiya nrsimha-pade magibo kakhana
nirapade navadvipe jugala-bhajana
bhaya bhaya paya yan'ra darsane se hari
prasanna hoibo kabe more daya kari
Weeping, I will beg at the lotus-feet of Lord Narasimha for the benediction of worshipping Radha and Krsna in Navadvipa, perfectly safe and free from all difficulties. When will this Lord Hari, whose terrible form strikes fear into fear itself, ever become pleased and show me His mercy?
yadyapi bhisana murti dusta-jiva-prati
prahladadi krsna-bhakta-jane bhadra ati
kabe va prasanna ho'ye sa krpa-vacane
nirbhaya karibe ei mudha akincane
Even though Lord Narasimha is terrifying toward the sinful souls, He offers great auspiciousness unto the devotees of Lord Krsna headed by Prahlada Maharaja. When will He be pleased to speak words of compassion unto me, a worthless fool, and thereby make me fearless?
svacchande baiso he vatsa sri-gauranga-dhame
jugala-bhajana hau rati hau name
mama bhakta-krpa-bale vighna jabe dura
suddha cite bhajo radha-krsna-rasa-pura
He will say, "Dear child! Sit down freely and live happily here in Sri Gauranga-dhama. May you nicely worship the Divine Couple, and may you develop loving attachment for Their Holy Names. By the mercy of My devotees, all obstacles are cast far away. With a purified heart, just perform the worship of Radha and Krsna, for such worship overflows with sweet nectar."
ei boli' kabe mora mastaka-upara
sviya sri-carana harse dharibe isvara
amani jugala-preme sattvika vikare
dharaya lutibo ami sri-nrsimha-dvare
Saying this, will that Lord delightedly place His own divine lotus-feet upon my head? I will experience sublime love for the Divine Couple Radha-Krsna and undergo the ecstatic transformations called sattvika. Falling on the ground, I will roll about at the door of Sri Narasimha's temple. (Srila Bhaktivinoda Thakura, Sri Navadvipa Bhava Taranga 36-40)
Bhaktivinoda Thakura prays to Sri Nrsimhadeva to clear his path to Radha Krsna worship. He wouldn't be praying to the equivalent of an "ordinary" Visnu tattva in Vaikuntha for entrance into Goloka.
Another argument against the identity of the two Nrsimhas is that the Pancaratra texts identify the Nrsimha avatara of the dasavatara as manifesting from Sankarsana: haris sankarsanamsena narasimha-vapur dharah (Padma tantra 1.2.31 and Visnu-dharmottara Purana 3.78(2).5-7) However, the vyuha Nrsimha is manifested from Pradyumna (CC Madhya 20.206).

Other Puranic narrations of the lila of Lord Nrsimha:

1. Nrsimha stuti from Garuda Purana describes a lila of Sri Nrsimha getting Siva out of trouble.

2. Puranic Encyclopedia: SARABHA VII. Virabhadra incarnated himself as Sarabha to defeat Narasimhamurti. (Siva Purana, Satarudrasamhita) and Narasimha then appeared as Gandabherunda to fight with him. [See also The position of tamasic sastras.]

3. As quoted by Srila Jiva Goswami in Bhakti Sandarbha, Anuccheda 106, the Visnu Dharmottara Purana describes a story of how Lord Nrsimhadeva once appeared from a Siva-linga to save a devotee named Visvaksena. Jiva Goswami says this Siva-linga is still present in South India and is known as "Linga-sphota".

Related:
Ahovalam—Revisited by Dhruva dasa
Ahovalam pilgrimage by Dhruva dasa (ITV video)
How Lord Narasimha came to ISKCON Sri Mayapur

15) Kurma - The Tortoise incarnation

He appeared to become the resting place of the Mandara Hill, used as a churning rod by the demons and demigods to produce nectar.

16) Dhanvantari - The father of Ayurveda

He appeared from the ocean of milk with the pot of nectar.

17) Mohini - The Lord's form of a beautiful woman

The Lord tricked the demons by appearing in the form of a beautiful woman and asked the demons for the nectar and distributed it to the demigods.

18) Vamanadeva - The dwarf incarnation, son of Kasyapa Muni and Aditi

Assuming the form of a dwarf brahmacari the Lord visited the fire sacrifice of Bali Maharaja. He begged three steps of land and with those steps He took the whole universe. He is the son of Kasyapa and Aditi.

19) Parasurama - Bhrgupati, the son of Jamadagni Muni and Renuka

He annihilated the ksatriyas twenty-one times because of their rebellion against the brahmanas.

20) Raghavendra - Lord Ramacandra, son of King Dasaratha and Kausalya

He came to perform pleasing work for the demigods, protect His devotees and kill Ravana and his followers. The Lord assumed the form of a human being and exhibited superhuman powers by controlling the Indian Ocean. His color is syama (dark blue) as per Ramayana Aranya khanda 17.7, 31.11, Sundara khanda 35.16, Yuddha khanda 28.18 and nava-dUrva dala-dyutiH (Laghu Bhagavatamrta 1.3.78). Bharata is dark as a raincloud and Sumitra's two sons are splendid as gold (bharato 'tra ghana-zyAmaH saumitrI kanaka-prabhau) (Laghu Bhagavatamrta 1.3.79).

Related:

Sri Rama Raksa stotra (sa, en, cz)
Rama Navami katha
Why was Sita exiled (as explained in Padma Purana)

21) Vyasadeva - The son of Parasara Muni and Satyavati

He divided the one Veda into several branches and sub-branches, seeing that the people in general were becoming less intelligent.

22) Lord Balarama - The first plenary expansion of the Lord

He acts as Lord Krsna's older brother in Their pastimes in Vrndavana.

namaste haladhRgrAma namaste muzalAyudha
namaste revatIkAnta namaste bhaktavatsala
namaste balinAM zreSTha namaste dharaNIdhara
pralambAre namaste 'stu trAhi mAM kRSNapUrvaja
(Brahma Purana 57.22-23, also Narada Purana 2.55.33-34 - var.: haladhRD nAme)

23) Lord Krsna - The original form of the Lord

Both Balarama and Krsna appeared in the family of Vrsni (Yadu dynasty) and in doing so removed the burden of the world.

24) Buddha - The son of Anjana

He appeared in the province of Gaya in the beginning of Kali-yuga to delude those who are inimical to the faithful. He also appeared to stop animal slaughter in the name of Vedic sacrifices by preaching non-violence and by not accepting Vedic authorities. There is also another Buddha incarnation mentioned in another Kali-yuga when Lord appeared to bewilder the demons who were destroying the inhabitants of other planets by flying unseen in space ships made by the demon Maya.

tataH kalau saMpravRtte haris tu sAMmohanArthaM cAsurANAM khagendra
nAmnA buddho kIkaTeSu prajAto veda pramANaM nirAkartum eva
(Garuda Purana 3.15.26)

25) Kalki - The son of brahmana Visnu-yasa

He will appear in the village Sambhala at the end of Kali-yuga. He will mount a horse, Devadatta, and taking His sword, will kill millions upon millions of degraded barbarian humans. He will also commence Satya-yuga. Details compilation from various sastras

Rupa Gosvami describes in his Laghu Bhagavatamrta 4 the 11 incarnations (other than the Purusa avataras) which have their abodes in the material world:

(1) Kurma: 53 Above that is the fifth place, a red world named Mahatala. In that place is a great lake 100 000 yojanas (800 000 miles) in breadth. There Lord Hari, in the form of Kurma, personally resides.

(2) Matsya: 54 Above that place is another of the same size, where there is a lake 300 000 yojanas (2 400 000 miles) in breadth. In that lake Lord Hari in His form of Matsya resides.

(3) Nara Narayana Risi: 55-57 Nara's friend Narayana Rsi resides at Badarikasrama.
(4) Nr-Varaha: Nr-Varaha is said to reside on Maharloka, where His abode is 3 000 000 yojanas (24 000 000 miles) in breadth.

(5) Ananta Sesa: Ananta Sesa's beautiful abode is 500 000 yojanas (4 000 000 miles) in breadth.

(6) Sveta Varaha: 58 The beautiful and self-effulgent realm Varahaloka touches the material universe as it stands beneath everything. There in the form of Svetavaraha, the Supreme Lord resides.

(7) Hayagriva: 59 Above that is another realm of the same size. This fourth realm, its ground coloured yellow, is named Gabhasthitala. There the Supreme Lord Visnu, splendid as hundreds of moons and decorated with golden ornaments, appears in a form with a horse's head.

(8) Prsnigarbha: 60 Above Brahmaloka is the realm of Lord Prsnigarbha.

(9) Vaikuntha: 63-64 In Svarloka is the home of Lord Visnu, who is the son of Vikuntha-devi. The Lord is also personally manifested in Vaikunthaloka.

(10) Ajita: 65-67 Lord Ajita's residence is considered to be on Dhruvaloka, and Lord Vamana's residence is considered to be on Bhuvarloka.< br/>

(11) Trivikrama: Lord Trivikrama's residence is said to be on Tapoloka, and Lord Narayana's spiritual asrama is on Brahmaloka. He has also created a residence above Brahmaloka. This was explained in Hari-vamsa (127.37), where Maharaja Indra said to Narada Muni:
68 O sage, above all the worlds Lord Visnu made His own abode, which eclipses even my realm.

They are Brahma (rajo-guna), Visnu (sattva-guna) and Siva (tamo-guna). Brahma is one of the living entities, but due to his devotional service he is very powerful. This primal living entity, master of the mode of material passion, is directly empowered by the Garbhodakasayi Visnu to create innumerable living entities. In Brahma-samhita (5.49) Brahma is likened to valuable jewels influenced by the rays of the sun, and the sun is likened to the Supreme Lord Garbhodakasayi Visnu. If in some kalpa there is no suitable living entity capable of acting in Brahma's capacity, Garbhodakasayi Visnu Himself manifests as Brahma and acts accordingly.

Similarly, by expanding Himself as Lord Siva, the Supreme Lord is engaged when there is a need to annihilate the universe. Lord Siva, in association with maya, has many forms, which are generally numbered at eleven. (They are not, technically speaking, avataras since this term is used only for Sri Visnu's manifestations. Hanuman is a partial expansion of Siva, rudra-kalodbhava, Brahmavaivarta Purana 4.62.62. He is not a jiva.) Lord Siva is not one of the living entities; he is a transformation of Krsna Himself. The example of milk and yogurt is given in this regard in Brahma-samhita - yogurt is a preparation of milk, but still yogurt cannot be used as milk.
Similarly, Lord Siva is an expansion of Krsna, but he cannot act fully as Krsna. The essential difference is that Lord Siva has a connection with material nature, but Visnu or Lord Krsna has nothing to do with material nature. In Srimad-Bhagavatam (10.88.3) it is stated that Lord Siva is a combination of three kinds of transformed consciousness known as vaikarika, taijasa and tamasa.

The Visnu incarnation, although master of the modes of goodness within each universe, is in no way in touch with the influence of material nature. Although Visnu is equal to Krsna, Krsna is the original source. Visnu is a part, but Krsna is the whole. This is the version given by Vedic literatures. In Brahma-samhita the example is given of an original candle which lights a second candle. Although both candles are of equal power, one is accepted as the original, and the other is said to be kindled from the original. The Visnu expansion is like the second candle. He is as powerful as Krsna, but the original Visnu is Krsna. Brahma and Lord Siva are obedient servants of the Supreme Lord, and the Supreme Lord as Visnu is an expansion of Krsna.

Manvantara-avataras

There are fourteen manvantara-avataras who appear during the fourteen ruling periods of Manus (manvantaras) in one day of Brahma (kalpa). They are unlimited in number. Each manvantara is presided over by different Indra.

Out of these fourteen manvantara-avataras, Yajna and Vamana are also lila-avataras. These fourteen manvantara-avataras are also known as vaibhava-avataras.

In SB 3.11.27 and 4.24.42 p. manvantara-avataras are confused with Manus as seen also from SB 1.3.5 p. Yajna is the manvantara-avatara in Svayambhuva Manu's reign, Vibhu is the manvantara-avatara in Svarocisa Manu's reign, and so on as per SB 8.13, CC 2.20.246 p., 2.20.319.

During each manvantara, in the Dvapara-yuga of each divya-yuga (maha-yuga), different sages accept the position of Vyasa or the compiler of the Vedas, Puranas, etc. In the Vaivasvata manvantara there were until now 28 Vyasas (Visnu Purana 3.3 - 3.4.5; this Purana is not spoken by Vyasa but by his father Parasara Muni: 1.1.26, 1.2.7-8, 3.4.3, 6.8.2-4,12):

In the next Dvapara-yuga the Vyasa will be Asvatthama, the son of Drona.

Yuga-avataras

The four yuga-avataras are also described in Srimad-Bhagavatam. In the Satya-yuga, the incarnation of God is white; in the Treta-yuga He is red; in the Dvapara-yuga, He is blackish; and in the Kali-yuga He is also blackish, but sometimes, in a special Kali-yuga, His color is yellowish as in the case of Caitanya Mahaprabhu.

This particular incarnation of the Supreme Personality of Godhead is foretold in Srimad-Bhagavatam (11.5.32):

kRSNa-varNam tviSAkRSNam
saGgopAGgastra-pArSadam
yajJaiH saGkIrtana-prAyair
yajanti hi sumedhasaH

"In the age of Kali the Lord incarnates as a devotee, yellowish in color, and is always chanting Hare Krsna, Hare Krsna, Krsna Krsna, Hare Hare/ Hare Rama, Hare Rama, Rama Rama, Hare Hare. Although He is Krsna, His complexion is not blackish like Krsna's in Dvapara-yuga but is golden. It is in Kali-yuga that the Lord engages in preaching love of Godhead through the sankirtana movement, and those living entities who are intelligent adopt this process of self-realization."

Colors of avataras

Annotation
In Sarva-samvadini 1 of Tattva-sandarbha, Jiva Gosvami elaborately comments on the verse which begins krsna-varnam, and adds: visnu-dharmottaradau yac ca dvapare suka-paksa-varnatvam kalau ca nila-ghana-varnatvam sruyate, “It is heard from Visnu-dharmottara and so on that in Dvapara the Lord has the color of a parrot’s wings, and that in Kali He has a dark blue color.” In Sanskrit, the attribution of names to colors can be confusing. The words syama and krsna are synonymous, but here the gloss 'nila-ghana’ can be interpreted as ‘blackish’ and as ‘dark blue’. Still, the word apita in the sense of syama is not seen in any scripture.

In regard to the colors of Avataras, there are many variations. For example, Sri Rama, an Avatara in Treta yuga, is said to be green like fresh durva grass (Laghu-bhagavatamrta 1.3.77). Elsewhere Sri Rama is said to be dark like a dark blue lotus petal: indivara-dala-syamam (Padma Purana 6.242.66).

Commenting on Laghu-bhagavatamrta, Baladeva Vidyabhusana writes: yat tu dvapare ’pi kvacit skande hari-vamse ca pitatvam uktam, tadapi kadacitkam astu, barer nanavataratvat, “Sometimes, yellowness in Dvapara too is mentioned in Skanda Purana and in Hari-vamsa. Still, let it be so now and then, for Hari has various Avataras” (Saranga-ranga-da 1.1.2). In the Kali yugas when Mahaprabhu does not descend, the Avataras are either dark blue (syama) or green like the color of a parrot’s wings: anyesu kalisu tu kvacic chyamatvena kvapi suka-patrabhatvena vavatarasyokteh (Saranga-ranga-da 1.1.2). At that time, those Avataras are mere empowered jivas (ibid.). In this way, the Lord’s designation as Triyuga is upheld, in consideration of the great majority. ("A Symphony of Commentaries on the Tenth Canto" to SB 10.8.13)

The "color of parrot's wings" ranges from green to dark green.

Saktyavesa-avataras

There is no limit to the saktyavesa-avataras. The saktyavesa incarnations are of two kinds - direct and indirect. When the Lord Himself comes, He is called saksat, or a direct saktyavesa-avatara, and when He empowers some living entity to represent Him that living entity is called an indirect or avesa incarnation.

Examples of direct or saksad-avataras are the Sesa incarnation and the Ananta incarnation. In Ananta the power for sustaining all planets is invested, and in the Sesa incarnation the power for serving the Supreme Lord is invested.

Examples of indirect avataras are (CC 2.20.369): the four Kumaras, Narada, Prthu (SB 4.19.3 p.) and Parasurama. Harinama Cintamani ch. 5 mentions also Buddha. These are actually living entities, but there is specific power given to them by the Supreme Personality of Godhead. When a specific opulence of the Supreme Lord is invested in specific entities, they are called avesa-avataras. The four Kumaras specifically represent the Supreme Lord's opulence of knowledge. Narada represents the devotional service of the Supreme Lord. Devotional service is also represented by Lord Caitanya, who is considered to be the full representation of devotional service. In Brahma the opulence of creative power is invested, and in King Prthu the power for maintaining the living entities is invested. Similarly, in Parasurama the power for killing evil elements is invested.

As far as vibhuti, or the special favor of the Supreme Personality of Godhead, is concerned, it is described in the Tenth Chapter of Bhagavad-gita that a living entity who appears to be especially powerful or beautiful should be known to be especially favored by the Supreme Lord.

"To answer your second question, you should know that Arjuna and Kunti Devi are not in Krishna Loka. They are eternally associated with Krishna only in the material world. Just like Krishna is always in the spiritual world, so similarly He is always in the material world and His Pastimes are going on there also. In the material world, Krishna also has eternal associates, such as Arjuna and Kunti Devi. There is a difference between the body and soul of Arjuna and Kunti Devi. But although Arjuna is with Krishna in innumerable different material universes at one time, still there is only one spirit soul who is Arjuna. This spirit soul expands into many different bodies and thus you can understand that there are also incarnations of devotees as well as incarnations of Krishna. This is the power of the spirit soul, that it is unlimited. Such conception cannot be understood while one is still in the conditioned state." (SPL to Saradia, Los Angeles, 12 December 1968)

"Tulasi plants are liberated souls who want to serve Krsna in that way. Anyone who even desires to serve Krsna is liberated, what to speak of one who is actually engaged in devotional service; expansion means remains in Goloka Vrndavana and at the same time expands all over the universe. Krsna can expand, so His devotee can also expand." (SPL to Ekayani, 31st August, 1971)

The mission of avataras and great devotees is to liberate all kinds of living beings, including those in rajas and tamas: SB 11.12.3-6, or Isvara samhita 8.175-7:

ekAntino mahAbhAgAH zathakopa-puras sarAH
kSonyAm krir-AvatArA ye lokojjIvana-hetunA
zAndilyAdyAz ca ye cAnye paJcarAtra-pravartakAH
prahlAdaz caiva sugrIvo vAyusUnur vibhUSanaH
ye cAnye sanakAdyAz ca paJcakAla-parAyanaH

Sathakopa and others, great devout saints, became incarnate on earth to save creatures. Sanaka and others, Sandilya and others, Prahlada, Sugriva, the Wind-Son (Hanuman), Vibhisana and others - the strict observers of the five-timed injunctions (pancakala) - these are the promulgators of the Pancaratra.

Recognizing the real avatara

When Lord Caitanya described the incarnation for this age of Kali, Sanatana Gosvami, who had been a government minister and was perfectly capable of drawing conclusions, directly asked of the Lord, "How can one understand the advent of an incarnation?" By the description of the incarnation for the Kali millennium, Sanatana Gosvami could understand that Lord Caitanya was indeed that incarnation of Krsna, and he could also understand that in the future there would be many people who would try to imitate Lord Caitanya.

"As one can understand the different incarnations for different millenniums by referring to Vedic literature," the Lord replied, "one can similarly understand who is actually the incarnation of Godhead in this age of Kali." In this way the Lord especially stressed reference to authoritative scriptures. In other words, one should not whimsically accept a person as an incarnation but should try to understand the characteristics of an incarnation by referring to scriptures. An incarnation of the Supreme Lord never declares Himself to be an incarnation, but His followers must ascertain who is an incarnation and who is a pretender by referring to authoritative scriptures. Real avatara does not want to be glorified (SB 4.15.23):

parokse 'smad-upasrutany alam
stotram apicya-vacah
uttamasloka-gunanuvade
na stavayanti sabhyah

O gentle reciters, offer such prayers in due course of time, when the qualities of which you have spoken actually manifest themselves in me. The gentle who offer prayers to the Supreme Personality of Godhead do not attribute such qualities to a human being, who does not actually have them.

Purport:

"Gentle devotees of the Supreme Personality of Godhead know perfectly well who is God and who is not. Nondevotee impersonalists, however, who have no idea what God is and who never offer prayers to the Supreme Personality of Godhead, are always interested in accepting a human being as God and offering such prayers to him. This is the difference between a devotee and a demon. Demons manufacture their own gods, or a demon himself claims to be God, following in the footsteps of Ravana and Hiranyakasipu. Although Prthu Maharaja was factually an incarnation of the Supreme Personality of Godhead, he rejected those praises because the qualities of the Supreme Person were not yet manifest in him. He wanted to stress that one who does not actually possess these qualities should not try to engage his followers and devotees in offering him glory for them, even though these qualities might be manifest in the future. If a man who does not factually possess the attributes of a great personality engages his followers in praising him with the expectation that such attributes will develop in the future, that sort of praise is actually an insult."

Any intelligent person can understand the characteristics of an avatara by understanding two features - the principal feature, called personality, and the marginal features. In the scriptures there are descriptions of the characteristics of the body and the activities of an incarnation, and the description of the body is the principal feature by which an incarnation can be identified. The activities of the incarnation are the marginal features. This is confirmed in the beginning of Srimad-Bhagavatam (1.1.1) where the features of an avatara are nicely described. In that verse, the two terms param and satyam are used, and Lord Caitanya indicates that these words reveal Krsna's principal feature. The other marginal features indicate that He taught Vedic knowledge to Brahma and incarnated as the purusa-avatara to create the cosmic manifestation. These are occasional features manifest for some special purposes. One should be able to understand and distinguish the principal and marginal features of an avatara. No one can declare himself an incarnation without referring to these two features. An intelligent man will not accept anyone as an avatara without studying the principal and marginal features. When Sanatana Gosvami tried to confirm Lord Caitanya's personal characteristics as being those of the incarnation of this age, Lord Caitanya Himself indirectly made the confirmation by simply saying, "Let us leave aside all these discussions and continue with a description of the saktyavesa-avataras."

Related:
Krisna and Radha, Goloka
Avatars in art

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