Matsya		Kurm		Varaha
In the Matsyavatara (Fish-incarnation), the Lord is said to have saved Manu (the progenitor of mankind) and the Saptarsis (the seven sages, mind-born sons of Brahma) along with their wives during the deluge. The world was repopulated through them later on.		Lord Visnu incarnated himself as the Kurma (the Tortoise) in order to support the mount Mandara which started sinking during the churning of the ocean (Samudramathana). The gods and the demons had jointly undertaken this adventure to get Amrta (nectar) from the ocean.		Next in the series is the Varahavatara (Boar-incarnation) in which the Lord killed the demon Hiranyaksa and lifted the earth out of the flood waters in which it had been submerged. This may be a symbolic representation of the extrication of the world from the deluge of sin by the power of the Supreme Being.
Narasimha		Vamana		Parashurama
When Prahlada, the great devotee of Visnu was being severely tortured by his father, the demon Hiranyakasipu, (who was a non-believer in the existence of an omnipresent and omnipotent God), Narasimha (Man-lion) appeared, emerging out of the pillar shown by him and killed him. Being a combination of man (the best of higher creatures) and lion (the best of lower creatures) Narasimha represents the best of creation. Incidentally this also proves the omnipresence of God. Narasimha is especially the embodiment of valour which is a divine attribute and hence worshipped by rulers and warriors. His Mantra is said to be very powerful, capable of destroying enemies and exorcising evil.		When Bali the grandson of Prahlada conquered the three worlds, Indra was deprived of his heavenly kingdom. At the request of Aditi, the mother of Indra, Lord Vigm incarnated as Vamana (the Dwarf), a young Brahmana boy, and approached Bali who was known for his munificence, for a gift of land that could be covered by three steps. With the first and the second he covered the earth and heaven, and with the third, he pushed down Bali to the netherworld. Hence he is also known as Trivikrama, one who encompassed the world with three big steps.		Parasurama (Rama with the battle-axe) is the sixth Avatara. Born as the son of the sage-couple, Jamadagni and Renuka, he exterminated the tyrannical among the Ksattriyas led by Kartavirya, who were oppressing the people. Whether this story has any historical basis and represents the struggle for supremacy between the Brahmanas and Ksattriyas, it is difficult to say. SrI Rama, the next incarnation, met Parasurama and absorbed his power into himself. Hence the latter is sometimes considered as avesavatara, an incarnation by the temporary possession of Visnu's powers.
Rama		Krishna & Balarama		Buddha
Sri Rama, one of the two most popular incarnations of the Lord Visnu, comes next in the series. His story is too well-known to need any repetition. He typifies the ideal man. His story, the Ramayana has now become an immortal epic. His name is known as the 'Taraka-mantra,' the Mantra that takes one across the ocean of transmigration.
Kalki		Nara Narayana

This list of the ten Avataras of Lord Visnu is by no means the standard one accepted by all. Taking SrI Krisna as Visnu Himself, he is not included in some lists. His place is taken over by Buddha. In some other lists, Buddha replaces Balarama. Ichnographically speaking, Buddha seems to have disappeared from such lists only after the 15th century.
Strangely enough, the purpose of the Buddha-incarnation was to mislead men of low birth and genius, who had become too proficient in the sacred lore and were a threat to the supermacy of the gods! This looks more like a joke than a serious proposition. It is obvious that the Hindus sealed the fate of Buddhism in this country by absorbing Buddha into the pantheon of the Avataras.
Harhsa, Satvata, Yajna, Dattatreya, Vedavyasa are some of the Avataras included in other lists, keeping the total as ten only. The number, however, rises sometimes to as high as twenty three.
CATURVYUHAS
Bhagavata or the Pancaratra religion, which preaches the cult of Visnu-Narayana, puts forth the theory that the Supreme Lord Visnu has four aspects of manifestation: (a) the Para or the supreme; (b) the Vyuha or the emanation; (c) the Vibhava or the incarnation and (d) the Arca or icon.
Para is the Supreme as He is, in all His glory. Vibhava represents the incarnations already dealt with. Arca is the descent of the Lord into the icon ceremonially installed and worshipped in the temples.
The Vyfihas or the emanations are four in number. Hence the term caturvyuhas, also called caturmurtis. They are: Vasudeva, Sankarsana, Pradyumna and Aniruddha. According to Vaisnava mythologies, while SrI Krisna is Vasudeva; his brother Balarama is Sankarsana. Pradyumna and Aniruddha are Krisna's son and grandson respectively. Historically speaking, it is possible that these Yadava heroes were, in course of time, apotheosised into these Vyuhas. Symbologically, Vasudeva represents Citta (mind-stuff), whereas Sankarsana stands for Ahankara (egoity), Pradyumna for Buddhi (intellect) and Aniruddha for Manas (mind). They represent the cosmic psychological evolution.
Later on, these Vyuhas were increased to as many as twenty-four. Iconographically, all these Vyuhas are identical in appearance except for the arrangement of the four emblems-Sankha, Cakra, Gada and Padma.
The Pancaratra theology often adds another aspect of the manifestations, viz., the Antaryamin (the indweller), which obviously, cannot be represented through icons.