What does John 13:24 mean?

The prior verse describes a disciple being directly next to Jesus (John 13:23). Peter wants to know which of the men present at dinner is the predicted betrayer Jesus mentioned earlier (John 13:21–22). It's interesting that Peter does not ask this question more openly—or, at least, that there is no record of him doing so. Given Peter's hot-headed nature, it might be that he wants a quiet reveal, so he can confront the person himself when given the chance.

Meals in this era were eaten in a reclined position, typically laying on the left side and eating with the right hand. The person Peter is speaking to is John himself, on Jesus' right, facing away. This is why, as the next verse shows, John will "lean back" in order to ask Jesus this question (John 13:25). This implies that John is asking in a hushed or personal way—possibly whispering. That position also explains why, though Jesus gives a sign, John does not seem to notice it being performed—he's probably facing the wrong direction (John 13:26).

This moment factors into what Jesus says to Peter later (John 13:38). No one suspects Judas, even when he leaves (John 13:28–30). When the disciples leave this room, it's possible their prime suspect is none other than Peter.