The Bhagavad Gita states the principle: whenever righteousness declines and unrighteousness rises, Vishnu takes birth age after age to protect the good, destroy the wicked and re-establish dharma. The Sanskrit word is avatara, a descent.
The standard list of ten, the dashavatara, is not the only one; the Bhagavata Purana gives twenty-two and calls the descents innumerable. The ten in the common sequence are Matsya the fish, who saves the first man and the seeds of life from a flood; Kurma the tortoise, who supports the mountain used to churn the ocean; Varaha the boar, who lifts the earth from the waters; Narasimha the man-lion, who kills a demon protected against death by man or beast, indoors or out, by day or night, and is therefore killed at dusk on a threshold by a creature that is neither; Vamana the dwarf, who asks for three paces of ground and covers the cosmos; Parashurama, the axe-bearing brahmin who destroys a corrupt warrior class; Rama, king of Ayodhya and hero of the Ramayana; Krishna, the most consequential of all; the Buddha, whose inclusion is variously explained; and Kalki, who is still to come, at the end of the present degenerate age, on a white horse.
Some Vaishnava traditions substitute Balarama, Krishna's brother, for the Buddha, and the treatment of the Buddha in the Puranas varies from respectful to polemical, sometimes presenting him as a deliberate deception to lead the unworthy astray. The sequence from aquatic to amphibian to animal to man-beast to human has often been noticed and occasionally pressed into service as a scheme of evolution; this is a modern reading, not a traditional one.