An Avatar is a picture used by someone to represent themselves online. It can range from anything such as a character from a story that someone likes, to a self-MoC, to a sports logo of someone's favorite team. Anything that represents someone in some way is a viable avatar.
On BZP, avatars are shown directly below the members name on the left side of a post in a topic. The avatar, being much smaller than a banner, typically does not have as much work go into it as a banner does, and therefore making avatars isn't the widely valued skill that banner making is. That said, there are still many popular av makers on BZP, and they have their own shops like banner makers.