[Lanka's answer][1] strikes me as comprehensive. Except for the etymology of the word. Pali *amata* and Sanskrit *amṛta* derive from the root √mṛ 'die'. The form here is a past participle used as adjective - *mṛta* means 'died, dead'; compounded with the 'a', *amṛta* means 'undying, deathless'. 

Ultimately it probably comes from Vedic as the Upaniṣads more frequently speak of the cyclic afterlife as *punarmṛta* 're-death' or *punramṛtyu* 'repeated dying'. It's this repeated death that makes the fact of cyclic existence ultimately unbearable and motivates the search for the deathless. And the opposite of repeated death is no death, the deathless. It can mean "immortality" in Vedic, in the sense that one's *ātman* exists permanently and once one transcends repeated death, then ātman rejoins and attains union with the universal principle of being, Brahman. Notably in the *Ariyapariyesana Sutta* it is Brahmā, the personification of Brahman, who urges the Buddha to teach after he opens the doors to the deathless. 

But as Lanka says, in Buddhism it is one of many synonyms for *nibbāna*.


  [1]: http://buddhism.stackexchange.com/a/10927/5715