Respect the double

When someone says "respect the double," as long as both players agree, then it means that both gamblers promise to allow the other person to "double-or-nothing" the gamble.

Thomas and Billy are gambling. Before they gamble, Billy says "respect the double" which Thomas agrees to. Thomas then loses and says "double." Since, Billy said "respect the double" before the gamble, he must allow Thomas to "double-or-nothing."