Avatāra
अवतारavatāraAvatāra means a coming down. In the Śrīvaiṣṇava tradition, which also calls it vibhava, the Lord comes down into the world without ceasing to be what He is, and takes birth in a form that suits the time and the need: the fish, the tortoise, the boar, the man-lion, the dwarf, Paraśurāma, Rāma, Balarāma, Kṛṣṇa and Kalki, who is told of as coming at the end of the present age. These ten are the descents counted most often; the lists differ in places, and the Śrīvaiṣṇava reckoning gives the eighth and ninth places to Balarāma and Kṛṣṇa. The tradition counts lesser descents as well, such as the sage Kapila, the Swan who taught Brahmā, and the twin sages Nara and Nārāyaṇa.
He comes, He Himself says in the Gītā, whenever dharma fails and wrong rises, to protect the good, to put down the wicked and to set dharma firm again. The teachers add that the body He takes is His own, taken by His own will and not made of the matter that binds ordinary beings, and that He comes for those who cannot reach Him by their own strength.
The hymn meets the descents in the deeds its names recall: the Bowman who bent His bow as Rāma, and the Bearer of the great mountain. Some names speak of them outright. He takes the forms of the avatāras and appears in the shape each devotee can love; He shows a clear form to those who love Him, in the descents and in the image in the temple; He dwells in body after body as the inner Self and comes again and again in His descents. The Śrīvaiṣṇava commentary hears whole runs of names as one descent or another, and counts the descent as one of the five ways in which the Lord is known, after His supreme form and the array.
SourcesDashavatara, Wikipedia (the lists of ten, and the Śrīvaiṣṇava list)Ramanuja list, SriPedia: the five forms of Nārāyaṇa (December 2002)Oppiliappan list, SriPedia: the five modes of the Lord (V. Sadagopan, 1997)