Here's what to look for:
RAM: at least 1GB, ideally 2GB.
Processor: Most will do. Ideally a Core Duo of some sort (just look for the words).
Hard drive: as much (in GB) as you can find. The more the better. You shouldn't get less than 100GB. Most PCs now will probably be around the 200GB range.
Try, if possible, to get a computer with XP and not Vista. Vista is a huge failure, and not worth the bother. XP works better.
If you're into more demanding games (the latest 3D stuff), then make sure you get more RAM and processor than usual, and also make sure it has a dedicated graphics card, and not just built-in.