Good question
It's mostly just the memory that you have to worry about, but for graphics-intensive games you need lots and lots of power - they usually print details of "minimum requirements" on the game box.

Most of the games I play are pre-owned older games (maybe 3-5 years old) which all run with no problems on my old AMD Athlon 1.3, and they're cheap - £2-£5 buys me a decent game.