At the end of the Aincrad arc of SAO on episode 14, on the big glass plane, Asuna was apologizing to Kirito and I don't know why. Was it because she jumped in front of the sword, or was it somthing else?
As you know Asuna and Kirito are madly in love with each other and that the game is deadly (quite literally). As any lover would do, they promised each other to survive the game and meet up in the real world, fall in love again and get married and so on. You know who he was fighting with but she thought that he was about to die from a lethal blow and decided to jump and save her man and eventually breaking the promise.
I have a few explanations for this.
Firstly, when Kirito and Asuna were on their honeymoon and they got called back, Kirito didn't allow Asuna to go for the boss battle, she told Kirito that her life would be meaningless if Kirito died. If Kirito really died, she said that she would commit sucide. Hence, when Kirito asked Healthcliff to prevent her from commiting suicide, Asuna knew instantly that Kirito knew that he didn't had a high chance of winning, so she decided to let herself get killed instead. At the end, she perhaps apologised for not believing in him.
Secondly, during the time when Asuna saved Kirito from the brink of death when Kirito was paralyzed and about to be killed by the Laughing Coffin, Kirito told her that his life was now in her hands, Asuna promised she would protect Kirito and she did. Soon afterwards, Kirito died and Asuna probably apologised for not being able to protect Kirito.
Thirdly, during the honeymoon, the both of them promised each other to meet up in the real world and go out together for real dates, love each other for real and also get married for real but the both of them broke their promises. Kirito broke the promise by asking Healthcliff to prevent Asuna from commiting suicide and Asuna broke it by shielding Kirito from Healthcliff's attack, thus die. So maybe Asuna apologised for breaking her promise.