Let's talk plainly for a second here (don't we always?). If you're running a game, then sometime, at some point, you're going to have to deal with missing a player at the session. Now, with luck, they'll have told you ahead of time but you don't always have such luck. Maybe there is a wedding going on, maybe a car broke down, maybe they stopped at a local Seven Eleven as it was being robbed and have to talk to the police for the next few hours. The point is, something has happened and one of your PCs is not going to be represented. So what do you do when that happens?
Well, luckily for you GMs have been handling this problem since RPGs got their start, and there are a number of ways you can deal with it. Below I'm going to detail the most common, along with some of their strengths and weaknesses.
Someone Else Controls Them
This, in my experience, was a lot more common back in the earlier days of gaming (at least, my earlier days). It is also something that is touched on in numerous comics such as Knights of the Dinner table. It works simply, Player A can not be here to control Character A, so Character A is handed to Player Y for the session. Player Y could be someone who was just going to sit in and watch, it could be another of the players, or it could be the GM themselves.
The benefit to this method is simple, the character is still around. They can still be in fights, still gain XP. The party isn't suddenly shy their mage, fighter, thief, or healer. Basically, the game can go on like it always does. The down side is that in more adversarial games, it can share information the player would rather leave hidden. Also, whomever is playing the character could be playing them wrong, or not know how to use the things on the sheet properly for best effect. You can also run into problems if the new controller gets the player killed, or gives away an item. All things you need to watch for, and all reasons why some players specifically ask to not have their character sheets ever handed to someone else.
The Eternal Leak
The other popular way, especially in dungeon crawls and other campaigns, is to say the character has gone off to take the Eternal Leak. Basically, it's a bathroom break, the character is always (safely) just behind the PCs, only they're otherwise indisposed. They dont get XP for encounters, or the session, but they aren't left behind somewhere and unless the party is wiped they're "safe".
The benefit to this is that you don't break up the party, the character is right there when the player does come back, and everything can go on as normal. The bad stuff comes in the form of lost XP, and the fact that the party is hindered without the abilities of one of their big characters. It's also sometimes hard to see how a PC on an eternal leak made it through certain scenarios, especially when the players who were there just barely made it through.
Doing Something Else
This one is simple, basically the PC has gone off to do something else while the player is away, and will rejoin the group when they're done. This one can work well, giving the player a chance at earning whatever XP they missed out on with some one on one time with the GM.
The strengths to this are fairly obvious, the character isn't with the group at all, they're gone and doing something else. You don't have to worry about someone else getting them killed, or how they got through things, they're simply gone. The downside is that this usually takes preparation and timing to do right, meaning it can't be done very often on short notice. Getting the player back to the group can also be awkward if they leave off in a cliff hanger, and the other player makes it back.
Those are the three most common ways I've seen GMs handle absentee players over the years. The "Doing Something Else" method usually works best, especially when it is planned in advance and works out as a plausible way for the character to be off and doing...something else. Beyond that, I tend to favor the Eternal Leak. I've never much cared for other players playing anyone's character but their own, it always feels and seems awkward and unnatural. At best, it'll be off putting. At worst, it can lead to a whole new level of griefing.
Of course, your mileage may vary, and I'm interested in hearing other ways to deal with an absentee player.