It depends on the sort of games you like really as well as your budget and your PC spec.
Platformer, arcade, turn based, first person shooter, racing, strategy, rpg, multiplayer, simulator?
I have a platform called Steam which acts as a games library and storefront. All the games have reviews, and you can sort by the sort of thing you're interested in, wishlist things, and keep an eye out for sales etc. Sites like Green Man Gaming will often sell a key for Steam - again with decent sales at times. Other places like GoG also have some great deals an allow you to download just teh game rather than being tied into a platform (like Steam)
Anyway, my wife quite liked Chicken Invaders which is £3.99 and will probably run on a potato. Audiosurf is quite a good, simple racer that builds a track based on the music on your PC. And cheap. One Finger Death punch is a lively beat-'em-up with relatively simple controls. Defense Grid Awakening is a nice tower defence game with a good few hours in it. Relseased in 2008 it will likely run on anything modern quite happily.