Two important factors that determine how high a basketball bounces are the force with which it is bounced and the height from which it is released. The higher you release it from and the more force you apply, the higher it will bounce.

Additionally, the height a basket ball (or any other ball) bounces has to do with the elasticity of the collision between the ball and the floor.

In an elastic collision, kinetic energy and momentum are conserved. In an inelastic collision, kinetic energy is not conserved. Energy can be lost to friction, sound, and heat.

When you drop a ball from a stationary position, it starts with all potential energy. As it falls, it gains speed and kinetic energy, but it still has the same amount of total energy. When it hits the floor in an inelastic collision, it loses some of its kinetic energy. As it bounces back upward, it exchanges its kinetic energy for potential energy, but it does not bounce as high as before because overall, it has less energy.

A collision between the basketball and floor will never be perfectly elastic (ideal). The closer a collision is to being perfectly elastic, the higher the ball will bounce.

So what makes the basketball behave more ideally? The less energy that it loses during the collision, the more kinetic energy will remain. One important factor is the type of rubber the ball is made out of. Rubber has two important properties – it is highly extensible (stretchy) and it returns to its original shape after deformation. When the rubber hits the ground, it is deformed; the side of the ball hitting the ground is squashed. When it returns to its original shape, the ball bounces back upward. The better the rubber is at snapping back to its original shape without losing energy, the higher the ball will bounce.

Another important factor is the air pressure inside the ball. When the ball is deformed, the volume inside of it changes and the air is compressed. The extra pressure from the compressed air helps it to return to its original shape and bounce up. If there is not much air in the ball to begin with and the pressure is low, the ball will not bounce well. However, if there is a lot of air, the pressure is high and gets even higher when the ball hits the floor. The compressed air expands and helps the ball quickly recover from the deformation and bounce higher. (If there is too much pressure, the basketball will explode).

The temperature of the ball will affect both the properties of the rubber and the pressure of the air inside the ball. Cooling a basketball will make it bounce less high.

For another explanation from a different perspective, this website may be helpful:

References:
http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html
http://en.wikipedia.org/wiki/Inelastic_collision McCrum, N. G., C. P. Buckley, and C. B. Bucknall. Principles of Polymer Engineering. Oxford: Oxford UP, 1997. Print.