go to ftp.idsoftware.com and you can find point releases for linux for all their games. Also there is source code for them. This is the easiest way to get the official Id software stuff for Linux. Quake1, Quake2, Quake3, The various Doom games including Doom3.
To use these games you need to have the real game media in the form of those game's pak files.
Personally I've been going oldschool with a modified quake2 engine game called Quetoo. Basicly for multiplayer quake2 action, very hardcore. (ah the memories)
Then there is Quake4, which may be aviable at Id's ftp sites. Unreal tournament 2003 and 2004. Wolfenstien, Enemy Terrirory.
Plus all the mods people have made for those games. Action Quake2, True combat, True combat: elite. Red Orchestra. Alien Swarm. SAS. All sorts of stuff.
Then there are independant and open source FPS. Most based on quake highly modified quake engines, but some on their own engine. Neziux, Code Red: Alien Arena. Cube. Fight Win Prevail. BZFlag (people tend to like this one a lot). Sauerbraten. and probably a half dozen more that I dont' remember off of the top of my head.
And all of these are just First Person Shooters.
Some game databases:
http://www.happypenguin.org/
http://www.icculus.org/lgfaq/gamelist.php
http://freshmeat.net/browse/80/
Most games for Linux are just small ones. There are some stuff that is pretty impressive though.
Keep a eye out for independent commercial game makers too, stuff like garagegames or linux game publishing has lots of cool stuff.