Sword Art Online's Kirito is capable of many things and has various abilities that other players don't. He was able to dual-wield weapons when it wasn't even in the coding for that particular world, keep Yui from being deleted by converting her information into environmental data, and even defy death. One of the more prominent feats that many people seem to be confused about is the ability to use Incarnation, what it is, and how it affects the user.
As YouTuber Gamerturn explains in their video on the subject, Incarnation is a term for when a user is able to override the limitations that the system has in place with nothing but sheer willpower. It's a concept seen more in the "Alicization" arc of the series, but it's also seen in the creator's other series Accel World.
The earliest example of what is later explained as Incarnation can be seen when Asuna is trying to reach Kirito before Kuradeel can kill him. Asuna's speed when trying to reach the valley where Kuradeel and Kirito are far exceeds what the system allows, although this isn't clarified in the anime and only explained more thoroughly in the light novels. Another time, she breaks the paralysis effect that Heathcliff sets on her in order to protect Kirito. Just after this, Kirito also uses Incarnation to briefly defy death. The concept of Incarnation isn't really expanded on until the "Alicization" portion of the story, but glimpses and hints are sprinkled here and there throughout the series.
One feature of Incarnation is that when activated by a user, their eyes will glow gold. This is an anime-only addition, most likely as more of a visual cue than an actual effect of using Incarnation. Without this visual cue, viewers wouldn't be able to tell whether someone is using Incarnation or just one of their regular abilities. One problem with this is that the animators seem to be inconsistent when characters use Incarnation. Kirito is the only one who is shown with golden eyes when doing so, even though other people have definitively also used it, and the reason behind this has yet to be explained. Asuna has used it more than once, Shiranami has likewise used it and at least seven other people have as well, but Kirito is the only one explicitly shown with golden eyes.
The answer could lie in the light novels, particularly SAO Progressive 006, when Kirito maxes out the Meditation skill to 500. By reaching this level of proficiency, it allows him access to a mod called "Awakening" which, as long as he keeps converting the same character he's been using since the beginning to new worlds, allows him to achieve a state similar to Incarnation. Asuna even points this out when he has golden eyes when using the mod. Incarnation allows Kirito to tap into past abilities and skills such as Awakening, and that could explain why his eyes turn golden -- by using Incarnation, he's able to use Awakening, giving him golden eyes.
However, this is only a theory, given that the golden eye detail came first in the anime, meaning that this explanation could have been added by the creator retroactively. As of the time of this writing, there has been no anime-canon explanation for the change in eye color.