This is an interesting question. There are a few factors determining how a ball will bounce, and each has a physical explanation and can be modeled at least partially using math. The way a ball bounces will primarily depend on the material the ball is made of, how inflated the ball is, and the angle that you bounce it.

Balls that are made of more elastic materials will be better than stiffer materials. Elastic materials are able to recover their original shape more easily than stiff materials are. If you were to watch the surface of a bouncy rubbery ball as it bounces (the kind people play four square with, for instance), you'll see the surface deform, or change shape, a little bit, and then recover its original shape as it moves away from the surface you bounced it on. Balls made out of stiff materials will not deform as readily.

The material that the ball is made of will also dictate how much friction the ball has when it encounters the surface you are trying to bounce it off of. This will largely come into play especially if you are bouncing the ball at an angle relative to the surface, rather than straight on. If you rotate/spin the ball as you throw it, it may bounce off the surface at a different angle than you would expect without the spin.

How inflated a ball comes into play also. If the ball is inflated very well, it will take less forced to bounce the ball as high in the air as it does to bounce a less inflated ball. A less inflated ball will actually absorb some of the energy that is used to bounce the ball by deforming the surface of the ball. The less inflated the ball is, the more you will see it to deform. But wait! We said that the ability to deform a little bit was a good thing. It is, but if a ball is too deflated, it will deform completely from its original shape and not be able to recover. Instead, the ball's surface would just conform to the shape of whatever surface you were trying to bounce it off of.

How well a ball bounces has a lot to do with what it is made of.The ball bouncing itself is demonstrating one of Newton's laws of physics--the conservation of momentum. When the ball falls to the ground, it has a certain speed and energy associated with it. When it bounces off the ground, some of its energy is transferred into the ground and some of its energy stays in the ball, and it bounces upwards.

Some of the energy in the ball goes to changing the shape of the ball (or deforming the ball) when it hits the ground. When a ball deforms a lot when it hits the ground, a lot of the energy is "spent" deforming the ball rather than in moving the ball upwards, so the ball doesn't bounce very well. It might help to think of a balloon filled with sand. When it hits the ground, it will flatten out, and usually won't bounce at all. On the other hand, a hard rubber ball won't deform much, and it can bounce very high. So, what the ball is made of and how much it deforms when it hits the ground are two important properties to describe the way the ball will bounce.

Not really a law of physics, no, but the materials that a ball can be made of will have properties that affect how it bounces. Generally, the more force the ball exerts when it is compressed will increase its bounciness. Balls behave like springs, basically.

When things bounce, the bounces can be either 'elastic' or 'inelastic.'

Hard metal balls bounce back without losing any energy. They make elastic bounces, or collisions. That's like in the 'Newton's Cradle' toy, where the 5 balls on strings bounce back and forth. When you pull back a ball on the end and let it go, the ball on the other end moves as far back as the distance you pulled the first ball back. Or, if the collisions were perfectly elastic, the distance would be exactly the same.

Rubbery balls have inelastic collisions, or bounces. If you drop a basketball, it doesn't bounce back as high as where you dropped it from. My granddaughter had some 'putty' or 'slime' that was surprising in how high it bounced. It was rubbery, of course, so it didn't bounce all the way up, but it didn't just stick to the floor.