How well will this computer run games?

You could get a desktop, custom, at hp. They have one called the hp8 i think that can support amd phenom II x6 1100t and a 3 gbs discrete graphics for around 1000 USd. ANd that can run anything really. For even more speed, i'd reccomend adding a ssd for boot and a ssd for the actual game files and a 2tb hdd for backup and other files since now most hp desktops custom can hold 3 slots for hard drives and ssds.