I agree with most of the above. I am going to assume from your statement about not having a great computer that you are not planning on doing a lot of multiplay so I will focus on single play.
Starcraft 2: Excellent game, but the campaign is not as replayable as I would like. I played it a lot for about a month and then was kinda done with it. The multiplay is great though.
Civ IV: Awesome. Like nearly all Civ games, infinitely replayable. Also, all the mods out there make for great, FREE, additional content. There is also a pretty good stand alone remake of Colonization that uses the Civ IV engine. I don't know if its nostalgia, the bug that let's me avoid the consequences of boycotting, or the great midi music, but I still think I prefer the 1995 super pixelated version of the game though.
Civ V: Haven't played it, but have heard great things post-patches.
Total War: This is really a great series, especially if you want to delve into a game for the next two months. I have only played the Rome titles though. There is a really good mod for it that makes the game WAY more historically accurate and strategically interesting (modified unit stats, cities maintain production of their cultural units, a larger map).
Hegemony Gold: This is an independent game I recently picked up and have really fallen in love with. It has several campaigns, the primary one being the conquests of Philip of Macedon. The engine is cool because you are playing a tactical game and a strategic game at the same time. Supply lines are super important, multiple battles can be going on at the same time, units are dependent on home cities to fill their ranks, battles incorporate morale and several tactics (flanking, charges), formations are very flexible but still simple (more so than Total war), the diplomacy system is simplified but surprisingly good, and food matters....A LOT. When I play Hegemony, it feels like a real military campaign because I think about things like, "How am I going to hold that city when it is so far away from my major food suppliers? The troops will starve and the city will rebel." and "I'll send my cavalry to take out that city's food supply so that I can starve out the garrison." Plus, I got it for $20. You should at least try out the trial version.