There are several ways to do it; team 1726 took the approach of using pneumatics to launch the ball. The cylinders were set up in such a way that they could build up pressure while the ball was on top of the catapult and then launch it over quickly after the pressure overcame the weight of the ball. (pictures
here)
39 also used a pneumatic launcher, but it works much better in our opinion. Instead of letting the cylinders build pressure, they just increased the airflow to the (single) cylinder to make it fire quicker. They had three valves running into it, allowing for a very simple and effective launcher (the best one I've seen so far).
The other main launching method I've seen involves using latex tubing to make a slingshot/catapult apparatus.
118 is noted for posting the design first, and many other teams have a similar set up.