Pardons, reprieves, and commutations are all forms of clemency.
Article II, Section II, Clause I of the Constitution says that the president has the "power to grant reprieves and pardons for offenses against the United States, except in cases of impeachment." The idea is that under a system of laws, there should be a way to forgive violations of those laws.
Clemency is kind of a kingly power. The framers felt like somebody in the federal government should have that ability to forgive federal crimes. They debated who it should be. They talked about sharing the clemency power between the president and the Senate, but they settled on the idea of it being vested in one person, so he could be held responsible by the electorate.
George Washington first used the clemency power; his most high-profile pardons came after the Whiskey Rebellion. He referenced the personal circumstances of the two individuals he pardoned, and the societal reasons for the pardons.
From FDR through Jimmy Carter, presidents had a clemency approval percentage—pardons and commutations divided by the number of applications—of about 25 percent. Ronald Reagan was at about 12 percent, George H. W. Bush at 5 percent, Bill Clinton at 6 percent, George W. Bush at 1.8 percent, and Barack Obama, so far, is at 2.6 percent.
Some of the difference has to do with the number of applications received. George H. W. Bush had less than 1,500, Clinton had almost 7,500, George W. Bush had 11,000, and Obama has more than 29,000 petitions for commutations alone, plus about 3,000 for pardons.
In recent years, clemency isn't that popular with the public, in part because of abuses of the power. The first example that comes to mind is the Iran-Contra pardons by George H. W. Bush. He had lost his reelection bid, and there was a chance that Casper Weinberger was going to be retried and might call Bush to testify. With Weinberger's pardon, Bush avoided potential embarrassment.
Clinton gave a bunch of pardons and commutations in his last hours as president, among them one to Marc Rich, a fugitive living in Switzerland. His ex-wife was campaigning to get him a pardon, a campaign that apparently involved a donation to the Clinton Presidential Library.
The president is often in a situation where he needs the approval of other branches of government. If he wants to enter a treaty he's got to get the Senate to ratify it. He's got to get Congress to declare war but clemency enables the president to check the decisions of the federal judiciary. This is a fairly unique ability. Although the framers had misgivings about a king, this is a potent power they thought the president should have regardless of its potential for abuse.