This article is from the Backgammon FAQ, by Mark Damish firstname.lastname@example.org with numerous contributions by others.
The introduction of the doubling cube into the game is largely
responsible for the leap in popularity of modern backgammon.
Each face of the doubling cube bears a number to record progressive
doubles and redoubles, starting with 2 and going on to 4, 8, 16, 32 &
64. At the commencement of play, the doubling cube rests on the bar,
between the two players, or at the side of the board. At any point
during the game, a player who thinks he is sufficiently ahead may,
when it is his turn to play and before he casts his dice, propose to
double the stake by turning the cube to 2. His opponent may decline to
accept the double, in which case he forfeits the game and loses 1
unit, or accept the double, in which case the game continues with the
stake at 2 units. The player who accepts the double now ``owns'' the
cube---which means that he has the option t redouble at any point
during the rest of the game, but his opponent (the original doubler)
may not. If, at a later stage he exercises this option, his opponent
is now faced with a similar choice. He may either decline the redouble
and so lose 2 units, or accept and play for 4, and he now ``owns'' the
cube. A player may double when he is on the bar even if his opponent
has a closed board and he cannot enter. Though he does not roll the
dice, for he cannot make a move, he still has the right to double.
Note that gammon doubles or backgammon triples the stake of the cube.