Lanka's answerLanka's answer 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 punarmṛtyu '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.