Generally, balls bounce. We want our basketballs to bounce, and our tennis balls and super balls too. There are some balls, like a hacky-sack ball or a medicine ball, that are specifically designed not to bounce. The question is, how do these so-called 'dead balls' not bounce?
To better understand, let's start by looking at what happens when a regular bouncy ball hits the floor. As the ball drops downward, it gains speed, and this speed is a kind of energy. As soon as it hits the floor, it slows down very fast, and the energy of its downward fall has to go somewhere. A bouncy ball momentarily stores most of that energy like a spring does. It deforms when it hits the floor, then springs back to its original shape, popping the ball back up again.
We can think of a ball's bounce in terms of energy: downward speed is converted into spring energy, then into motion again as the ball goes up. For a ball not to bounce, it needs to get rid of all that energy the first time it hits the floor, so there won't be any left to spring it back up again.
Dead balls do this with friction inside the ball. When a dead ball hits the floor, the materials inside it rub against each other as the ball deforms. The friction of this internal rubbing changes most of the downward energy into heat, not spring energy. After a session of playing hacky sack, put the ball to your cheek. It should feel warmer.