Avatar

An avatar is a sort of self-insert that acts as the author's representation in a story or fanfic. The word comes from the Sanskrit avatara, which was used for Hindu gods and goddesses that manifested on Earth on purpose and had greater powers than normal humans. Compared to normal self-inserts, and sometimes even Mary Sues, avatar characters are aware that they are in a story, can, and often will give themselves special powers, and can twist the canon far worse.

Alternately, you might mean the small icons used to represent people on internet forums and blogs.

You might also be looking for Avatar: The Last Airbender.