An avatar is someone who carries out the will of a deity, spirit or similar being in the mortal world. Traditionally, an entity needs an avatar because it cannot interact directly with the "real" world - due to being extraplanar, too powerful, impossible to comprehend, or the like - and thus creates a unique creature, or chooses an existing creature it deems worthy, to act on its behalf. Avatars are usually imbued with the power of their creators/benefactors, though sometimes an avatar may not even be aware of the true source of his or her powers.
Usually avatars are mystical in nature and serve a distinct being with a consciousness, such as a god. However, embodiments of abstract concepts, emotions, or a collective conscious - such as "the avatar of hope" or "the avatar of the homeless" - may also fit this category.
While "avatar" is also often used to describe a certain player's signature or "main" character (or a character who is based on a player's real-life self), this category isn't intended to correspond with that sense of the word.