Half Decent PC that can run some games?

I'll admit I'm a complete moron when it comes to computers. I have no idea what anything is/does, and I've been using the same old ancient rig for years now and it is slowly dying. I need a new PC.

I don't have the time/money/desire to build myself a dedicated gaming PC, even though I know that is most likely the best option. I'm not really in the market for a gaming PC per se, although I would enjoy being able to run some games that I'd buy (or have already bought) from Steam. My budget is around 600$ mac (tax included), and I really just want a fast desktop that can run average games at mid-to-low settings, even if it means locking every game to 30FPS. Is there any pre built PC from any company that can suit these needs or is this unrealistic? If I'm being too vague feel free to let me know and I'll specify things more clearly if I need to. I just want to make a smart choice because money is tight right now, and who better to ask than the smartest gaming community on the internet?