A chess clock is actually comprised of two seperate clocks with a button or switch on each side. They are either digital or analog.

After you make your move, you press the button on your side of the clock. This stops the clock on your side and starts the one on the opposite side. Your opponent is now free to mull over his move while his clock ticks..

The digital ones are quite costly, and can run for about $100. The analog ones can be found for about $40-$60.

See Also: Chess - Time Controls