A Basketball ball is dropped onto the floor.What force causes the ball to bounce?

When a basket ball is dropped and hits the floor the surface wall of the ball contorts. This causes the air inside the ball to pressurise. The pressure of the air pushes back at the surface wall causing the ball to bounce back up.

 

Here is a good link explaining the process further,

 

"The more air pressure a basketball has inside it, the less its surface will bend or deform during a bounce, and the more its original energy will be stored in the compressed air inside. Air stores and returns energy more efficiently than the material that the ball is made from. If the ball is underinflated, some of its energy is wasted in deforming the ball as it bounces, and the ball will not rebound very high. For the most elastic collision possible between the ball and the floor, you want a highly pressurized ball. (But you knew that already, didn't you! Now you know why!)"

 

http://www.worsleyschool.net/science/files/physicsof/basketball.html