Real-time strategy, collecting resources, building towns, and fighting. I do like video games set in the past, but I would be interested in this kind of game set in any period.
Okay. The RTS par excellence is considered by most RTS fans to be Starcraft. It's an old game by Blizzard (1998 or so) so it should work on everyone's computer. It's one of my favourites as well. It'd be a good idea to play the first one before moving on to SC II, which comes out next year (theoretically).
Other excellent RTS games....
-Command & Conquer: Red Alert 3, set in an alternate universe where the Soviet Union is ridiculously powerful and the Japanese Empire never fell
-Company of Heroes, a typical Axis vs. Allies title
-Rise of Nations, a game that's sort of like Age of Empires but spans across all of human history