Sword Art Online's first season found Kuradeel attacking Asuna. In defending her, leading man Kirito's hand gets sliced off by Kuradeel's blade during the confrontation. A few moments later in a different setting, Kirito appears to be holding a teacup and saucer with both hands, so what happened?
In Episode 10 of Sword Art Online, Crimson Killing Intent, Kirito endures a training session with Godfree and Kuradeel on the 55th Floor as a newly-appointed member of the elite Knights of the Blood Oath. Godfree and Kirito are soon poisoned and paralyzed by Kuradeel during this training, which enabled Kuradeel to murder Godree with little to no struggle. After uncovering some betraying news about Kuradeel, Kirito is painfully stabbed by him in the leg. Asuna comes running to Kirito's defense, but as Kuradeel attacks her, Kirito springs to her defence and ends up getting his hand cut off by Kuradeel's sword amidst the rush of chaos. In a heroic display of drive and determination, Kirito ultimately kills Kuradeel using the Embracer attack.
The Aftermath
After Kirito and Asuna emotionally declare their love for one another having survived this traumatic ordeal, there is a change in the scene. The lovebirds are having dinner and a spot of tea on the 61st Floor in Selmburg after agreeing to spend the night together. Kirito is comfortably holding his cup and saucer - a two-handed task - as he spends a quiet evening with Asuna. This begs the question as to when and how exactly did his hand grow back.
Sword Art Online is a video game, therefore losing health is not uncommon. Generally speaking, gaming characters' health decreases by a variety of means to give the game and characters realism. Usually when these characters enter a town, home, or in-keep of some kind, this indicates that means to boost the characters' health are available and thus a way for the character to be healed. A possible interpretation of Kirito's hand regrowth is that when he entered the town again, his health increased, and thus the debuff had expired and his hand grew back.
How Did it Happen?
Even if reentering Selmburg did not automatically heal him, the possibility of Kirito's hand regrowing before arriving is entirely plausible due to the three-minute rule established by the light novels. In the shot where Kirito's hand is chopped off, his life gauge in the top left corner of the screen shows an icon that indicates a body part loss. These losses are a status effect or "debuff". According to the Sword Art Online light novels, body part losses regenerate in three minutes. This explains why Kirito is able to use both his hands in the next scene, three minutes definitely would have passed by then.
While the unexplained regrowth of Kirito's hand is sudden, this is basically how standard video games operate. The logic is simple: get hurt, get healed. The alternative of a character wandering the rest of the game on the brink of death is not enjoyable or practical to anyone, therefore healing in busy modern places makes sense, as it indicates the presence of medicine and doctors who would normally heal wounds. This assumption exhibited in the anime is apt to stay true to this element of gaming structure.
On the other hand, the anime presented Kirito's regeneration as open and random. Not everyone is familiar with how basic gaming operations, therefore they may see this regrowth as a plot hole or lazy writing, perhaps. It can be frustrating or confusing to see a sudden unexplained change, especially coming off such a thrilling and fast-paced turn of events just witnessed. As a result, many would have questioned the regeneration of Kirito's hand with annoyance.