With some variations in different incarnations, after the lawyer prosecuting mobster Sal Maroni Harvey Dent; (Harvey Kent?) is splashed in the face with acid and becomes Two-Face, they take Maroni's lucky double-headed coin. Most famously, Harvey Dent uses a two-headed silver dollar, one side of which is scarred, the other side clean. Flipping the coin dictates Dent when making any significant life decisions. If the coin lands on the scarred side, Two-Face will pursue acts of evil. If the coin lands on the unmarked side, he is compelled to commit acts of good. At one point Dent as Two-Face was so dependent on the coin for any of his criminal acts that he could be easily defeated by taking the coin away from him or preventing him from seeing the result of the flip. Harvey Kent was usually shown at this level of dependency. Dent has since lessened his dependency on the coin flip to whether a situation should be implemented, rather than each individual act.

Trivia

In the movie Batman Forever and Batman The Animated Series, Batman easily defeated Two-Face by throwing up multiple quarters and silver dollars while the villain was in the middle of flipping his coin. The former resulted in death of that universe's Two-Face
In The Dark Knight, Dent was burned in a fire instead of hit with acid or scarred by Maroni. Following his updated origins, one side of the coin was scorched instead of scarred.