The most obvious benefits of fixed gear bikes are that they are relatively light due to their mechanical simplicity (of course this depends on the frame that you have...) and that they are low maintenance.
But there are other benefits that might not be as obvious to people who have never ridden a fixie: For example, the fact that your feet are connected directly to the rear wheel gives you a lot of feedback on traction, which can be a big advantage in slippery conditions. The lack of a freewheel is also said to encourage a more effective pedaling style, and to increase endurance.
Another benefit is the ability to slow down using the pedals. But I don't recommend taking this too far!
While you can brake and decelerate on a fixed gear using just the pedal, we do not recommend ever using a brake-less fixed gear bike. I know that it's considered "cool" and "hardcore" in some circles, but it's not worth it. In some situations, nothing can replace good old brakes.