"The buck stops here" is a phrase that was popularized by U.S. President Harry S. Truman, who kept a sign with that phrase on his desk in the Oval Office. (Footage from Jimmy Carter's "malaise speech" shows the sign still on the desk during Carter's administration.) The phrase refers to "passing the buck," i.e., handing responsibility to someone else, and the fact that the president has to make the decisions and accept the ultimate responsibility for those decisions.
The buck, a marker or counter, was used to indicate the person whose turn it was to deal in the game of poker. If the player did not wish to deal he could pass the responsibility by passing the "buck", as the counter came to be called, to the next player.