Balls, including ping pong balls, bounce after they hit a solid surface because an equal and opposite force from the surface pushes them back upward. Ping pong balls have a tendency to bounce well because they're made out of a plastic material that is especially stiff and springy.
With the exception of energy lost during the bounce itself, the force pushing the ball back upward is equal and opposite to the force with which the ball originally hit the ground. When a ball is squishy or easy to deform -- think of a partially deflated basketball -- more energy is lost during the bounce.
All balls deform to some degree when they hit the ground. This can be seen in high-speed photography that shows that balls are flattened for a split second after they hit a surface. This deformation results in a loss of energy. Instead of all of the energy pushing the ball back upward, some is spent deforming the ball, and then pushing it back to its original shape. Squishier balls do not bounce as well as more rigid balls because they expend excessive energy regaining their shape. Alternatively, ping pong balls bounce very well due to the stiff plastic material from which they are made. The stiffness restricts deformation, so less energy is needed to restore the ball's shape.