A fixed gear bicycle is defined primarily by the lack of a freewheel, which is a device like a ratcheting wrench with one cog or several cogs of various sizes around it. This device allows the cogs, when pulled from the top by the chain, to rotate the wheel clockwise (if viewing the bike from its right side), but the cogs can rotate counter-clockwise freely of the rest of the wheel, and the wheel can rotate clockwise freely of the cog. This allows the bike to coast -- the bike can continue to roll forward without the pedals turning.

A fixed gear bike cannot coast, which can slow you on the downhill to depending on your cadence (how fast you can turn the cranks). A fixed gear bike also, almost by definition, does not have multiple gears. While theoritically possible, even ultra-modern high-end deraillurs (the devices used to move the chain from one cog to another) can occasionally, if only momentarily, bind up. With a freewheel this is no problem, but if the cog stops on a fixed gear the back wheel stops, which is potentially disasterous. I know a guy who has a fixed gear with multiple chainwheels, and I think he's asking for it, since the front deraillur is even more unreliable than the rear.

There are several advantages to a fixed gear. Of course, having fewer parts (one cog on each end of the drive drain, no bearings or pawls in a freewheel, no deraillurs chain tensioners) means less weight, which means less force to move and an easier ride. The lack of a deraillur also means that the drive system is direct, which drastically increases the efficiency of translating your pedalling to movement of the bike. The least obvious by probably most appealing advantage is that you can feel the traction between the rear tire and the road in your legs, which makes a fixed gear bike much safer on slippery surfaces like snow and ice.

Another pretty neat thing is that you keep your bike at a stop without putting your foot down! All you do is turn the front wheel to the side to stabilize bike and rider left to right, then slightly allow the rear wheel to move forward or backward in response to slight fluctuations in your balance, like a unicyclist. This makes toe-clips and even clipless pedals a lot easier to deal with, since you don't have to put any mental energy into re-engaging with the pedals. It's an all-around much more direct and more intimate experience of bike-riding.