1. Open Arena - It is perhaps the most popular among all free, open source FPS games.
2. Alien Arena - Another popular game based on the source code of Quake II and III. This is a graphics intensive game and requires pretty hi end graphics card for a smooth performance.
3. Nexuiz - This is a multiplayer (online) FPS game which can be played in single player mode as well. The game boasts of realistic gaming.
4. Sauerbraten - This is an extremely fast FPS game with exquisite graphics. Package includes a map editor.
5. Assault Cube - Without a doubt the most "fun" game of the lot. Exhaustive multiplayer support lends an amazing depth to the gameplay. The file size of the game is also small, around 50MB which should leave you with no excuse whatsoever for not giving it a try. A small warning though, there are literally gaming freaks playing this game, so if you're starting out with FPS, this isnt the game for you.
6. Warsow - Last, but not the least. Warsow. This is seriously one of the more "beautiful" FPS games out there. Strangely beautiful graphics with amazing gameplay makes this a must play.
This was of course a small list. Other significant games include Adanaxis (a space based FPS. Does a game get cooler that that?), Tremulous, Enemy Lines, etc.
Every single game mentioned here is available in the Ubuntu Software Center. Simply search for it and install.
(Suggestion: Search for "Tilda" in the Ubuntu Software Center. You will be pleasantly surprised. Hint: Its a terminal emulator with FPS theme)