To pass the buck is to shift responsibility off yourself and onto someone else. It may mean giving a dirty job to some one else, or simply shifting the blame.
The phrase originated in poker games in the American West. In the 1800s cowboys and the like would mark the rotation of the office of dealer by passing around an object; today we call this the dealer's button, or simply a button. But back when it had no name, people used whatever term came to hand. Often, a knife was used to mark the dealer (no, not like that), and as these knives often had hilts made from the horns of a buck, these markers were eventually simply called bucks. A buck could serve other purposes as well;
"The 'buck' is any inanimate object, usually knife or pencil, which is thrown into a jack pot and temporarily taken by the winner of the pot. Whenever the deal reaches the holder of the 'buck', a new jack pot must be made."
-- Draw Poker by J.W. Keller, 1887
So passing the buck was a normal and necessary part of a poker game. Some people suppose that the phrase may have picked up its pejorative connotation because one could prematurely pass the buck if one did not want to take on the responsibilities it implied. While this was fine if he was simply ducking out of being the dealer, dodging the responsibility of contributing to a new jackpot was no doubt a more serious concern. However it came about, the current usage was in general use by the early 1900s.
Harry S. Truman's popular catchphrase, "The buck stops here", also originated from the poker buck; a friend of Truman's saw a sign with this phrase while visiting a Federal Reformatory, and related it to the president. No doubt it was intended to inform the inmates that they would now have to take responsibility for their actions, but it struck the president as a sign that he should take responsibility for everyone's actions. There is also a theory that the slang term 'buck', meaning an American dollar, might have come about once silver dollars became a popular dealer's button; this is much less certain, and the chronology is a bit off (the two terms originated about the same time, but the 'dollar' usage appeared slightly earlier than the 'poker' usage). It is more likely that use of the word buck came form the Native American's use of a buckskin as a monetary unit (An 'Indian's dollar', so to speak).