First, the power is only granted to the President to grant pardons and reprieves for Federal crimes.
“The President shall…have power to grant reprieves and pardons for offenses against the United States, except in cases of impeachment.”
A reprieve is granted under particular circumstances. For example, if they discover new evidence or testimony that might have affected the case’s outcome if it had been known during the trial, the President can grant a reprieve to allow the new evidence to be examined.
The pardon, however, is a little different. It doesn’t mean the person is innocent, but that the sentence might have been unjust. The pardoning power has been called “the conscience of the nation.” It’s supposed to be exercised when a true sense of justice calls for intervention. Alexander Hamilton wrote the following about the pardoning powers of the President in Federalist #84.
“He is also to be authorized to grant ‘reprieves and pardons for offenses against the United States, EXCEPT IN CASES OF IMPEACHMENT.’ Humanity and good policy conspire to dictate, that the benign prerogative of pardoning should be as little as possible fettered or embarrassed. The criminal code of every country partakes so much of necessary severity, that without an easy access to exceptions in favor of unfortunate guilt, justice would wear a countenance too sanguinary and cruel…”
James Iredell of North Carolina said,
“Another power that he has is to grant pardons, except in cases of impeachment. I believe it is the sense of a great part of America, that this power should be exercised by their governors. It is in several states on the same footing that it is here. It is the genius of a republican government that the laws should be rigidly executed, without the influence of favor or ill-will–that, when a man commits a crime, however powerful he or his friends may be, yet he should be punished for it; and, on the other hand, though he should be universally hated by his country, his real guilt alone, as to the particular charge, is to operate against him. This strict and scrupulous observance of justice is proper in all governments; but it is particularly indispensable in a republican one, because, in such a government, the law is superior to every man, and no man is superior to another. But, though this general principle be unquestionable, surely there is no gentleman in the committee who is not aware that there ought to be exceptions to it; because there may be many instances where, though a man offends against the letter of the law, yet peculiar circumstances in his case may entitle him to mercy…This power, however, only refers to offences against the United States, and not against particular states.”