Okay, so I've seen quite a few threads on good open source games (having made two myself), and decided to compile a list of them. Anyone who notes that I've missed one or more, please provide a name, short (and accurate) description, as well as a link.
Here's the beginning-
Vega Strike- an absolutely awesome Free-Form style Space Sim, that has some campaign/story elements, if you choose the right path. Search for http://vegastrike.sourceforge.net and you should find it.
FreeOrion- still under development, I found myself enjoying this open-source TBS 4X game. Search for FreeOrion, you should be able to find it very easily. The Research subjects are interesting, and its in a playable state as well. The only thing I really don't like is that it takes REALLY LONG TIMES to do, well, anything. Find it at http://freeorion.sourceforge.net.
Warzone 2100- originally a commercial title RTS, but released as open source, I love this one a lot. Researching advanced weapons technologies, materials tech, and hull/propulsion/production designs is pretty neat. The best part though is that it has an IN-GAME unit designer. Choose from different hulls, propulsion systems, and weapons/support turrets. Googleing it should turn up the proper results. Once again, SourceForge is the place to get it (I think, may be wrong, so GOOGLE!!)
Star Wraith: Shadows of Orion- while not actually open-source, the publisher, Star Wraith 3D Games, has decided to release this as free download (no adware in it, I've played it a good amount, trust me). The Campaign is pretty nice, and so is the mode where you simply run or create a mission. Overall, an excellent older game. Get it from the official SW3DG site, under the 'Other Games' or 'Downloads' pages. One or the other, forget which. A space-sim, like VS.
Star Wraith IV: Reviction- the sequel to SW:SoO, it has also been released for free. Pretty much the same thing, but with a new campaign (unsure, Campaign won't work for me). Same site.
RiftSpace- an SP-only free-form space sim. Same site as the other SW3DG titles.
The Ur-Quan Masters- the freeware open-source released version of Star Control II. I'm pretty sure everyone knows what that is.
PiArmada- a space strategy combat TBS game, it is based in the Vega Strike universe (I believe, as it won't work for me ATM). Open-source; it should be obtainable from the Vega Strike forums.
Thousand Parsec- a TBS 4X-style game, it is being converted to the Vega Strike universe. It can run with SP, but is intended as an MMO-style game.
There may be others, I don't know all of them.
EDIT: Added several more; I am unsure of the links though Google (or Bing if that floats your boat) should fill in there-
Spring- an RTS engine/game based on Total Annihilation. Different 'mod' packs for different games.
Battle For Wesnoth- fantasy-based TBS game.
FreeCiv- based on Sid Meier's Civilizations, an MP TBS workstation/PC game.