As of right now, we only know the stories of the following Avatars:
Wan (We were told his story in Legend of Korra: Book II)
Kuruk (We know he was a Water-Bending Avatar, and the spirit Koh ate the face of the woman Kuruk loved)
Kyoshi (We know how she took down Chin The Conqueror and formed the Dai Li)
Roku (Two whole episodes were dedicated to his story in Avatar The Last Airbender: Book III)
(the next two are obvious)
Aang
and Korra