The free FPS world seems to be largely focused on multiplayer deathmatch games, in the tradition of Quake 3: Arena. For example, a while back I wrote about a game called OpenArena (an open content replacement for Q3).
Recently, maybe more for procrastination than anything else, I've been giving Cube 2: Sauerbraten a spin. Unlike OpenArena, but like many other free games, it's not true Free Software, since, while the engine is Open Source, the actual game content is not. It is still free to download, though, for several platforms, and also available in Debian's repositories.
It's basically a fast paced, futuristic deathmatch game in a rather similar vein as Quake 3. The emphasis is very much on multiplayer, and there's plenty of modes available, from traditional deathmatch to various capture the flag variations. But it does have some entertainment to offer for a loner like me, as well. (Online multiplayer games really aren't my thing. For one thing, I suck at them. And I don't have the kind of dedication you'd need to really get into them.)
In addition to bots in regular multiplayer modes, there's a 'DMSP' mode, where you basically just have to wipe out a bunch of monsters in DM maps. Then there's a regular, progress oriented single player mode. There's hardly a lot of depth, just tons of monsters, and the occasional door or switch you activate simply by going near it. There's only a handful of single player maps, I'm afraid, from various contributors, and mostly without any real plot. The quality is pretty varied, some just kill-everything-that-moves sort of things, while a few might feature more thoughtful level design and attempts at more involving scripting. All in all, there's enough for at least a few light play sessions in them. Of course long, involved campaigns aren't necessarily always called for. I wonder how many times I've played just the first few levels of Doom, just for the fun of it...
One interesting feature is the 'slow motion' mode. In this you don't die, but the game slows down considerably when you're low on health, while you slowly regenerate. This is simultaneously helpful and an annoyance. It's quite fun, actually.
Technically it's not exactly state of the art, of course, but looks fairly good for a freeware game, and there are some pretty nice water and explosion effects, etc. And it runs mostly fine on my current notebook (a fairly new, but reasonably cheap model).
So, for me, as a quick, light Doom replacement (because that's the game I obviously compare all light FPS games to), Sauerbraten is not too bad.