As was mentioned, what kinds of games do you like?
--I'd start with Team Fortress 2, as it is free-to-play. It's a cartoonish shooter that's a lot of fun. Other good FPS titles are Counter-Strike: Global Offensive and Left 4 Dead 2 (if you like zombies; the first game's really shallow and redundant, but the second's a bit better).
--If you like League of Legends, I think I still have a bunch of DotA 2 beta keys, and I could send you one.
--If you like tower defense games, I suggest Plants Vs. Zombies (I'm sure you know what it is by now, haha) and Sanctum. Sanctum is something of a FPS tower defense. You get guns in an FPS setting to shoot the waves of enemies, but you also get to build defenses to protect your "core." It's in a grid setup, so you try to position your defenses on the grid to create a long maze for ground troops to get through. It's a really cool game.
--Music games like Audiosurf are nice. You probably know what it is, but in case you don't: you have a track with colored blocks to collect. The track varies by the beat of your music.
--If you like the Tycoon games, Roller Coaster Tycoon 3 is a good one.
Those are some I can throw out there quickly, but I play mostly on Xbox. Playing PC games on your bed (where I use my desktop from) is a pain, so I don't usually bother. Let us know which types of games you like, and we can probably give better examples. Of course, just browsing on Steam is really easy, and most games have demos, it seems.