Newton's Third Law
The usual summation of Newton's Third Law of Motion states "For every action there is an equal and opposite reaction." When two objects collide, the direction of the force on one object is also applied in the opposite direction on the second object.
Stationary Object
To understand how a bouncing ball operates, first consider a stationary ball. The ball rests on the ground, neither rolling nor bouncing. Gravity exerts a downward force on the ball proportional to its mass, and according to Newton's Third Law, an opposing force pushes the ball upward. Since the two forces are equal, the ball stays still, neither sinking into the ground nor lifting into the air.
Falling Ball
In the case of a falling ball, the force equation no longer stays in equilibrium. At the moment of impact with the ground, the ball exerts a force on the ground equal to its mass multiplied by its acceleration. Since the ball is no longer at rest, the force is greater. In turn, a greater force presses the ball upward.
Bounce
The greater upward force overcomes the ball's mass, careening it back into the air. The higher the acceleration of the ball, the greater the opposing force. For this reason, the harder you throw a ball against the ground, the higher it bounces.
Material
Bounce also depends on the material of the ball. Consider a ball made of steel or wood. On impact, the opposing force dissipates without bouncing. In order for a bounce to take place, the ball must be elastic enough to conserve the energy. More efficient materials, such as rubber, retain the energy better, allowing for higher bounces.