You may be looking for those small user icons found on message boards, blogs, and forums.
In fiction, an Author Avatar is a character that acts as the author's representation in a story or Fanfiction.
An avatar is different from a self-insert. A self-insert is meant to be a copy or a stand-in for the author him/herself, but an avatar is merely there to represent the author rather than literally be the author's body.
If this seems confusing, bear this in mind: an author avatar is often meant to express the author's perspective without actually including a character that is supposed to contain the author's identity. No matter how divorced from the real author, the purpose of a self-insert is to contain the author's identity. An avatar is more like the author's puppet or mouthpiece than their mask.
The word 'avatar' 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.
Avatars are not inherently bad — in fact, many good characters in fiction are based on some aspect of the author. Being an author avatar is not a flaw in of itself, but changes the meaning and increases the gravity of other charges when they do arise. For example, a character that expresses the author's opinions may not be bad in concept, but it definitely is bad if this character exists ONLY to fulfill the author's shipping wishes as a matchmaker!
Most, if not all, Mary Sues have some element of an avatar to them, if they are not Self-Inserts. Sometimes this is intentional, because the Sue is meant to insert the author's opinions into a continuum. Sometimes this is unintentional, because many Mary Sues are not aware that they are actually just spouting the opinions of the writer rather than being their own developed person.
In a worst case scenario, Mary Sue avatar characters are aware that they are in a story, and will begin to change the canon as they see fit. They can, and often will give themselves special powers beyond what a Mary Sue trying to 'fit in' would dare to carry.