The bikes used in cycleball are fixed-gear bikes. On most bikes, there is a flywheel that allows you to stop pedaling while you are moving and coast, but with a fixed-gear, the pedals are chained together with the rear wheel in such a way that the pedals always move whenever the rear wheel is moving. This is important in cycleball for a couple of reasons. First, unlike a conventional bike, it is possible to go backwards simply by pedaling backwards. Second, a fixed-gear allows the rider to hold the back wheel still by keeping the pedals still so he can balance the bike better while stopped, similar to what a unicyclist does.
Besides the fixed-gear, there are a few other things that are unique to a cycleball bike:
The saddle is located above and behind the rear wheel axle, allowing the rider to take all the weight off the front wheel while he is "kicking." Riders spend most of their time up on their legs, off the saddle, but usually briefly put their weight down on this rear saddle when kicking.
Up-turned handlebars allow the rider to pull back to raise the front wheel in the air. They also give the rider greater control and more power to kick the ball with the front wheel.
Brakes are not needed, because the bikes usually aren't moving very fast and the bikes can be slowed down by the fixed-gear pedals. Hand brakes would just get in the way and not be used.
You can see all of these in action on a video: