God Vishnu incarnates on Earth from time to time to eradicate evil forces, to restore the dharma and to liberate the worthy ones or devotees from the cycle of births and deaths. Vishnu in his full avatar as Krishna speaks in the Bhagavad Gita, Chapter 4 Shloka 8: "For the deliverance of the good, for the destruction of the evil-doers, for the enthroning of the Right, I am born from age to age."
To know more, please refer to chapter 15 of Essays on the Gita viz. The Possibility and Purpose of Avatarhood. The other purpose for divine incarnation is to make possible on earth the furtherance of evolution itself.
You could also read the 1000 names of Vishnu as recited by Bhishma in the Mahabharatha.
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wiki:Dashavatara
source:www.aurobindo.ru/workings/other/van_vrekhem-beyond_man.htm
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Mathsyam (Fish)
Vishnu takes the form of a fish to save Manu from the deluge.
Kurmam (Tortoise)
When the devas and asuras were churning the Ocean of milk in order to get amrita, the nectar of immortality, the mount Mandara they were using as the churning staff started to sink and Vishnu took the form of a tortoise to bear the weight of the mountain.
Varaha (Boar)
He appeared to defeat Hiranyaksha, a demon who had taken the Earth, or Prithvi, and carried it to the bottom of what is described as the cosmic ocean in the story.
Narasimhan
Vishnu descended as an anthropomorphic incarnation, with the body of a man and head and claws of a lion. He then disembowels the rakshasa at the courtyard threshold of his house, at dusk, with his claws, while he lay on his thighs.
Vamanan
The dwarf who revealed himself as Vishnu to King Bali, covering the heavens with his first stride, the netherworld with his second, and then when Bali offered his head, placed his head on it, granting the king immortality and making him ruler of the netherworld.
Rama
The exemplar supreme, whose story is told in the sanskrit epic, the Ramayana
Ram is your soul. Sita is your heart. Ravan is your mind that steals your heart from your soul. Lakshman is your consciousness, always with you and active on your behalf. Hanuman is your intuition and courage that helps retrieve your heart to re-animate your soul
Parasuraman
The first Brahmin-Kshatriya in Hinduism, or warrior-saint, with duties between a Brahmana and a Kshatriya. He is a Chiranjivi (immortal), and believed to be alive today in penance at Mahendragiri.
Balarama
He may have originated in Vedic times as a deity of agriculture and fertility, hence the name Balaram (Rama with a plough).
Krishna
The preceptor (teacher) supreme, who reveals his true form (vishwarupam) to Arjuna while expounding the Bhagawad Gita
Buddha
The founder of buddhism. King Siddhartha who renounced his kingdom in search of the eternal truth. In an alternate list of the dasavatharam, he displaces Balarama.
Kalki
The yet-to-come avatar of Vishnu, said to come before the establishment of the Kingdom of Heaven, contrary to what Georges says.
Ganesha
Vishnu