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Whenever righteousness wanes
and unrighteousness increases
I send myself forth. In order to protect the good
and punish the wicked,
In order to make a firm foundation
for righteousness,
I come into being age after age.

- Bhagavad Gita

In Hinduism, avatar (or avatara) is a word used to describe the descent of a deity into manifest form. The term is derived from ava (down) and tṝ (to cross), and has come to signify the being who descends rather than the act of descent - the original meaning. Avatars are aspects of the Divine Self who incarnate for a specific purpose. There are many avatars of the Hindu Goddess (Shri Devi or Shri Shakti). The aspect of the Divine Self as guru, or teacher, also incarnates. Perhaps most of the Hindu avatars are incarnations of the deity Shri Vishnu, who is the aspect of the Hindu Trinity responsible for maintaining the universe and its laws. The other two aspects of the Hindu Trinity are Shri Brahma (creation) and Shri Shiva (dissolution). Avatars of Shri Brahma are very rare. Shri Shiva is thought not to have ever incarnated, as He represents the unmanifest Self. There are, however some great rishis and saints who are considered to be partial incarnations of Lord Shiva, for example Shri Ramana Maharshi (no connection to TM). Some of the aspects of Shri Shiva are believed to have incarnated, such as Shri Bhairava. The Shiva Puranas tell of a Shiva avatar in the form of a bird-lion, called S(h)arabha, who incarnated in order to quell the exessive rage of Shri Narasimha, the man-lion avatar of Shri Vishnu, after He killed the demon Hiranyakashipu. In some versions of the myth, a powerful and wrathful bird, with two heads, emerged from Narasimha and fought with Sharabha for many days until Shri Narasimha's anger was satisfied. The double-headed bird can be found on the emblem of Karnataka state in India, and also in European/Russian heraldry and Amerindian tradition.

Double-headed eagle of the Russian royal emblem.