I've heard of letting the GM or someone else play the absent player's character, but that feels awkward (for lack of a better word) to me, so I prefer not to do it in my games.
For a while, I had the absent player's character go along with the rest of the group and just sort of be there, but that got awkward a number of times, when the PC was supposed to remember something that happened while their player was absent.
My current method is to come up with an in-game reason for the absent player's character to be somewhere apart from the rest of the group. It sometimes takes some storytelling gymnastics, but it seems to work for us.
While I'm on the subject, how do you handle characters whose players have left the game (for whatever reason)? I usually have them fade into the background for a while, then leave permanently. I've heard of some groups who give an "abandoned" character to someone else to play, but I've never seen that work out well, so I prefer not to do it myself.