It's a game, so we only ever play with the PC's that have available players at the table - same as if, say, only 4 of your 5 poker buddies turned up, you'd only deal 5 hands (not 6), right?
Generally, we hand-wave the PC absence, and their return, as quickly and painlessly as possible - usually that's the DM's call, sometimes player(s) might help try and find ways to explain it "in game"; they found something else that needs their attention, or whatever.
But really, no-one cares much how it happens, it's just accepted by me and my buddies (me as DM, and another guy who DM's a game I play in), from many years of experience, that it's way better to just leave that PC out of the session, than try and include them.
There's nothing much to be gained by trying to keep that PC involved, the story will survive and the remaining players will just have to make do. In fact often being one short, leads to better role-play due to less pressure for spotlight time, and/or better combat choices due to people needing to pay attention more and make the best of what they have. Some of my most memorable sessions have been when there are only 3 of our 4 or 5 listed players available at the table. Occasionally, as DM, I've included a simple NPC for each player to run as part of an important scene where there's only 3 PC's, but usually we just run with 3 PC's and it works out fine enough - the players love it.
On the flip-side, there is much to be lost by trying to include the missing PC - in order to try and "preserve the story", you end up making all sorts of ridiculous things happen - you get a special DM-PC who could easily be overly favoured and/or played terribly, or if a player runs them you risk them playing that PC badly and getting them killed and/or other PC's because the player is overloaded, or you just end up compromising the story in other ways, all for what, "continuity"? It's a game, not a TV series, movie, book etc.
I never dock XP for missing a session, everyone is always at the same XP level in my games because no-one misses a session on purpose, just like no-one dies on purpose and no-one in my games gets punished for deciding to swap out a character; my DM runs this differently, but as the person who's usually got the most XP and is often a level above the others, I personally disagree with his logic, I would actually rather that all of us were at the same XP level as my PC.