Meaning
Avatāra (Sanskrit: अवतार — from ava-tṛ, 'to descend across') means the Lord's voluntary descent into embodied form within the material world. Unlike the souls who are compelled into bodies by karma, the Lord's avatāras are entirely voluntary (svecchā) — taken for specific cosmic and salvific purposes.
Why the Lord Descends
The Bhagavad Gīṭā (4.7-8) gives the classic statement: 'Whenever dharma declines and adharma rises, I manifest Myself. For the protection of the good, the destruction of the wicked, and the establishment of dharma, I take birth age after age.'
Śrī Vaiṣṇava ācāryas add a deeper reason: the Lord descends out of overwhelming love for His devotees — to be accessible, to be seen, to be touched, and to offer grace. This is why the Lord's avatāra body is aprākṛta (non-material, transcendental) — not a karma-bound body but a divine manifestation of His will and love.
The Ten Primary Avatāras (Daśāvatāra)
The classic list: Matsya (fish), Kūrma (tortoise), Varāha (boar), Narasiṃha (lion-man), Vāmana (dwarf), Paraśurāma, Rāma, Balarāma (or Kṛṣṇa), Kṛṣṇa (or Buddha), and Kalki (future). The Āzhvārs composed many mangalāśāsanam hymns celebrating the Lord's various avatāras.
Vibhava Avatāra
In Śrī Vaiṣṇava theology, avatāras are classified under vibhava — the Lord's manifestation through specific, time-bounded descents — as distinct from His para (transcendent) form in Vaikuṇṭham, His vyūha (cosmic) forms, His vibhava forms (avatāras), His antaryāmi (indwelling) form, and His archa (image) form in temples.