Asuna was one of the 10,000 players trapped in Sword Art Online, where she later was the sub-leader of the Knights of the Blood Oath guild. Her skills with the rapier had her earned her the nickname Lightning Flash.
Asuna is a kind and helpful young woman who cannot abandon another in trouble and takes the game very seriously but is determined to clear it before Kirito tells her to enjoy SAO a bit. Asuna is also somewhat proud and despite her kind personality wont hesitate to get physical with those that challenge her authority or make fun of her abilities, such as cooking. She is not afraid to act out on her own and will take matters into her own hands from time to time.
However Asuna is also someone who can be controlled and manipulated by her emotions. She even recklessly attacked The Gleam Eyes after seeing the Aincrad Liberation Force players getting slaughtered because she can't handle seeing others die in front of her, this is also shown during the final duel with Heathcliff on the 75th Floor and jumped in front of Kirito to took an attack that would have otherwise killed him.
In ALO, Asuna's personality had not changed a bit except that she was not as proud as she was in SAO and her personality began to develop further still after meeting Yuuki in the events of Mother's Rosario. Seeing how Yuuki still moves forward despite knowing her death is inevitable Asuna takes after her and resolves to keep moving forward no matter what happens.