That is a good physics question! Yes, gravity does affect they way balls bounce. Gravity pulls the ball toward the ground, slowing the ball down so that each bounce is shorter and shorter, until eventually the ball stops bouncing. The force of the ball hitting the hard ground puts an equal force back onto the ball, causing it to bounce up. This happens because balls are made out of an elastic material like rubber, which means they can be dented or stretched and then return to their normal shape (think of what happens when you squeeze a balloon). If the ball was made of something softer like silly putty, it would just be squashed on the ground and wouldn't bounce. If the ball was made of something harder like glass, it would hit the ground and break.

Try bouncing balls with different stiffness (like a fully-inflated soccer ball compared to a half-inflated soccer ball, or a tennis ball compared to a baseball) and see what happens. Also, the type of surface you bounce the ball on will affect how high the ball bounces - try bouncing a ball on hard concrete and then on soft grass. Can you guess what would happen?

Note from a reader:
In reference to Answer #1 above, "If the ball was made of something softer like silly putty, it would just be squashed on the ground and wouldn't bounce."

As every child knows, Silly Putty bounces! It's one of the unique features of this material.

Probably better to go with clay or Play-dough as an example.