They were first used extensively in tournament chess, and are often referred to as "chess clocks", but their use has since been adopted for tournament Scrabble, Shogi, Go, and nearly every competitive two-player board game. A game clock ensures that neither player overly delays the game.
The simplest time control is "sudden death", in which players must make a predetermined number of moves in a certain amount of time or forfeit immediately. A particularly popular variant in informal play is "speed" or "blitz" chess, in which each player is given five minutes on the clock for the entire game.
The players may take more or less time over any individual move, which means that whilst the start of the game is a relatively quick succession of moves, the players can spend a large period of time thinking over the more complex moves later. The players are permitted to leave the table during their opponents turn--but there is nothing stopping their opponent from deciding on their move and pressing the button to restart the opponent's clock.
Analog clocks (as pictured on the right) are equipped with a "flag" that falls to indicate the exact moment the player's time has expired. Unfortunately, additional time can't easily be added for more complex time controls, especially those which call for an increment or delay on every move, such as some forms of byoyomi. Therefore more versatile digital clocks are becoming increasingly popular.
Digital clocks and Internet gaming have spurred a wave of experimentation with more varied and complex time controls than the traditional standards. It remains to be seen whether new standards will emerge, or a greater variety of time controls will be widely used.