I have to second the mentions of OpenTTD, Battle for Wesnoth, Ur-Quan Masters and Nethack.

I'd also like to add:

• Widelands: A Settlers 2 clone.
• Chromium B.S.U.: A top-down scrolling shooter. Don't let any enemies pass. Perfect if you need 5 minutes of adrenaline.
• Scorched 3D: It's Scorched Earth, but 3D.
• Frozen Bubble: Hard to describe. It's a bit like Dr. Mario.
• GL-117: 3D air combat. The graphics are "a bit" dated, but the game is a lot of fun still.
• Kobo Deluxe: A 2D action-puzzler. It doesn't have the most stunning graphics, but it sure is fun to play.

Then there are also some open-source re-implementations of commercial games (that need the original game files) that haven't been mentioned yet:

• CorsixTH: Theme Hospital.
• OpenMW: Morrowind
• OpenRA: Red Alert (and others)
• ScummVM: Various point&click adventure games (Monkey Island, Legend of Kyrandia, Loom,....)
• Return to the Roots: Settlers 2
• VCMI: Heroes of Might and Magic 3