Nexuiz is a fun open source FPS based around a heavily modified Quake 1 engine -- and when I say heavily modified, I mean "Looks like an early Xbox 360 game, instead of a PC game from 1996." It's kind of like a cross between
Quake and
Unreal Tournament, and it can be a lot of fun.
Quake Live should go without saying. If you've been living under a rock for the last several years, it's a free version of
Quake III that runs in a browser window.
There's also
Open Arena, an open source version of
Quake III, but I'd say
Quake Live makes it essentially obsolete.
If flight sims are more your style,
Privateer: Gemini Gold might be right up your alley. It's an opensource remake of
Wing Commander Privateer, complete with all of the cutscenes and voiceclips from the original game. I never really got into it, but that's because I prefer games like the main series
Wing Commander ones; it's a pretty solid
Elite clone, if you're into them.
I know there's a bunch more I'm forgetting, but this post is getting long enough as it is; every second that passes increases my chances of getting ninja'd.