The Battle for Wesnoth is a Free, turn-based tactical strategy game with a high fantasy theme, featuring both single-player, and online/hotseat multiplayer combat. Fight a desperate battle to reclaim the throne of Wesnoth, or take hand in any number of other adventures. Easily modifiable. You can even download other users' add-ons from the main menu.
Wesnoth is an incredible strategy game with great production values in graphics and music, especially considering it's an open-source project. Some of the built-in campaigns are really top-notch. Multiplayer is also a lot of fun. Modding possibilities are, in a word, fascinating.
The only minor flaws are really in the difficulty levels and balancing. Newbies should experience tons of frustration. Despite all of its great sides, there's just *something* minor lacking in it that would keep me glued to it.
Still, I'd definitely say it's an unmissable game that everyone should try.