See Residual self image.
An avatar in the Matrix series is the virtual representation of a person within the Matrix. A person's avatar is based on the subconscious image they hold of themselves; hence Neo, upon first jacking into the Matrix after his Extraction, has a full head of hair, not the shaved cut he has in the real world. Clothes can also be altered to a person's liking, or based on the programming of the operator monitoring them.
This only applies to Redpills, as they are conscious of the nature of the Matrix and jack in of their own accord. Bluepills use this programming as well, though this is only so their "bodies" in the Matrix are accepted by them. Redpills also cannot change their appearance while jacked in, unless they change it by conventional means within the Matrix itself (such as a haircut or physically changing their clothes).
One's avatar is controlled by her or his mind. Electrical impulses are sent from their brain to their virtual bodies, not their real bodies. Movement is therefore as easy in the Matrix as the real world. However, avatars are able to perform a much wider range of motions and abilities that cannot be experienced in the real world. Leaping across buildings, for example, can be achieved in the Matrix both because of a lack of a real body and that the laws of physics in the Matrix can be affected by willpower.
The death of a person's avatar results in the death of the person it represents (with markedly few exceptions) as it literally kills their mind (or, at the very least, severs it's connection to their body). As Morpheus puts it, "the body cannot live without the mind". Similarly, injuries received to the avatar within the Matrix are transfered to the body, as "your mind makes it real". However, this only applies to internal injuries. For example, Mouse's body, after Mouse is hit by multiple bullets in the Matrix, coughs up blood, though no bullet holes appear on his body.
A person's mind can only leave their avatar and return to their body via a hard line exit. When this happens, the avatar itself is removed from the Matrix, dispersing into lines of code.
The avatar appears in the Zion Archives. An avatar in Hinduism is the incarnation or physical manifestation of an immortal being. The term is frequently used in computer games, Internet forums and virtual reality as an icon representing a user or player.