Impeachment, the decision that the President or other high official has acted in a way incompatible with his continuance in office, is beyond doubt the most powerful of the checks and balances, and as such, is yet another indication that the authors of the Constitution had greater confidence in the legislative branch, in which power is distributed among a large number of people, than in the executive, in which power is concentrated in a single individual. It is worth noticing that there is no counterpart power: the Congress can remove the President from office, but the President cannot dissolve Congress.
The Constitution provides that President (as well as members of the judiciary) can be removed from office in cases when he (there have been no women Presidents yet...) is found to have committed acts that are incompatible with the fulfillment of his duties.
In terms of procedure, the House brings charges: the offenses for which a President or judge can be impeached are "Treason, Bribery, or other high Crimes and Misdemeanors." The actual trial takes place in the Senate, with the Chief Justice of the Supreme Court presiding. Conviction, and thus removal from office, requires a two-thirds majority.
Two Presidents have been impeached: Andrew Johnson (in 1868, just after the Civil War) and Bill Clinton (in 1998-99, in the so-called "Monicagate" scandal). Richard Nixon resigned before impeachment proceedings were completed (in 1974, in the Watergate scandal [which, perhaps obviously, is the origin of the use of the suffix "-gate" to mean a scandal. "Monicagate" is just one example]). No President has ever been convicted of impeachment charges by the Senate, so none has ever been removed from office.
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The impeachment trial of Andrew Johnson was, in purely legal terms, based on violations of the law that the President was alleged to have committed. However, it is clear in retrospect, and was so even at the time, that the main, not to say exclusive, reason for the attempt to remove Johnson from office was the deeply rooted disagreement between the President and Congress about what policies to adopt to recreate the United States after the Civil War. In other words, the impeachment was motivated largely by political differences, much more than by specific violations of the law. Johnson's acquittal (by a single vote) was assumed for generations to have established the precedent that impeachment should not be motivated by political considerations. The interrupted impeachment of Richard Nixon conformed to this principle. As noted above, Nixon resigned before he could be impeached, but he would almost certainly have been removed from office, as there is no question that he abused his Presidential power.
Bill Clinton's impeachment trial was, it became increasingly evident, motivated essentially by political bitterness. It eventually became clear that Clinton had, indeed, had sexual relations with Monica Lewinski, and furthermore that he had lied in claiming he had not, but it also became clear that his sex life had not interfered with, or been an abuse of, his duties as President. The nation spent months of time and millions of dollars following the unfolding events, even after it became clear that Clinton would be acquitted and that the acquittal was completely justified.