You *could* try Halo: CE. That's both Combat Evolved and/or Custom Edition (the standalone moddable multiplayer version) (I really don't want people to slug me for this but I suppose it'll happen eventually)
Granted, the single-player is old news, and everything looks like textured carpet on a low resolution, but everything is quite runnable unless your computer is actually a cardboard box wired to leads made of plastic embedded in a 10-year-old shit-coated lemon.
The multiplayer is usually the drawing point of the game - low pings are so easily achievable I live in Australia and it feels like some random guy from Mexico has his computer hooked up next to mine in a LAN game, along with his other random friends from Siberia, Korea, the Netherlands and France next door to my house. Drawbacks is that a portion of these players will inevitably be a bunch of pricks, but if you find a server where the people play properly then it's a strangely addicting multiplayer.
Gameplay can be easily gotten used to, there are plenty of game modes to play around with, mechanics are only *slightly* broken, and once you get sick of the base game you can move into Custom Edition territory, which can then be backed up with Halo 2 and Halo 3 emulated content. Although the modding takes quite a while for a game of its generation, tags and their silly stupid implementations makes releasing custom content sometimes as slow as molasses at sub-zero. Theoretically its moddability knows no bounds.
People have mentioned UT04, and you could also try UT: GOTY edition, if you like that sort of pacing for your games. It's been modded to hell and back, it's fast, simple, learning curve isn't too steep (just takes practice to use it well obviously) and there's volumes of stuff to choose from when you want to customise. I've gotten tired of that sort of game, but I used to be into it and when I was, it was pretty much the only FPS I really wanted to play.
Being an older generation title, I would confidently take a guess and say that your computer should probably have no problem running it at all.
You could also try some of the Wolfenstein titles. Those are great fun (just make sure you turn your brain off when it comes to the storyline) and the multiplayer title Enemy Territory is also great, although I find some of the maps and modes to be slightly limited in scope. (As a side note, what did you guys think of the Wolfenstein title previewed at E3?)
I still like Star Wars Jedi Knight 3: Jedi Academy, and play it every now and then. Granted, that's a third-person-perspective you get when chopping shit up with a lightsaber, and the first person element isn't really that inspiring, but that multiplayer is also really great and because it was using the Quake III engine (although probably expanded and used to a far greater extent than most people would have ever expected), the game has also been modded for the good part of this decade since Jedi Knight II.
Those are just my picks, but there's so many out there to choose from. I haven't mentioned some of the more recent titles, but it's just because everyone's pretty much mentioned all of them and you said that your computer wasn't all that flash.