What is a Fixed Gear Bike?
A fixed gear bicycle doesn't have a freewheel (don't worry, they're not all as weird-looking as the one in the picture that illustrates this post - I just couldn't resist that funny-looking basket). This means that when the rear wheel turns, the pedals turn. The rear sprocket is screwed directly into the rear wheel's hub, so there's no way around it.
This means that, for example, you can't just coast down a hill without moving you feet. You have to turn the pedals at the same rate as the rear wheel is turning at all times! This isn't a minor difference, it's a big deal. Some people hate it, but some cyclists swear by it (especially bike messengers).