Rubber is made from long-chain polymer molecules. When you hold the ball in your hand, these long molecules are tangled together like a ball of molecular spaghetti. Atomic motions within the rubber molecules then return them toward their original, tangled shape. Much of the energy of the ball's downward motion becomes upward motion as the ball returns to its original shape and bounces into the air. The energy in the ball that isn't converted into motion becomes warmth.
Rubber balls are elastic because they return to their original shape.
The organized motion of the falling ball becomes an organized deformation of the rubber of the ball, which then becomes an organized motion of the bounced ball. On the other hand, rubber polymers can be made in which the molecules move more freely, rub together more, and turn organized motion into disorganized vibration. The ball will hardly bounce. Instead, it gets warm.