"Therefore, I shall resign the presidency effective at noon tomorrow. Vice President Ford will be sworn in as President at that hour in this office."
- President Richard Nixon, in the Oval Office at the White House, on August 8, 1974
Impeachment alone doesn't remove presidents ...
In the entire history of the United States, only two presidents have ever been impeached, which were Andrew Johnson and Bill Clinton. Richard Nixon was never actually impeached by either house of Congress. He was, however, credibly threatened with it during the Watergate scandal, and thus forced to resign in this way. It takes both houses of Congress, you see, to remove a president from office. Thus, when the Senate refused to remove Andrew Johnson and Bill Clinton during their impeachment trials, the fact that they had been impeached by the House of Representatives to start this process had only a symbolic value. They had to be convicted by the Senate after this to be actually removed from office.
... although one president resigned under the threat of impeachment (and subsequent removal)
It is thus one of the ironies of presidential history that both of the impeachments of an American president in our history failed to remove their intended target, while the only president ever to be forced out of office did so only under threat of it. Besides making the point about how both houses of Congress have to be on board with this to pull off a successful removal (as with Nixon), there is one other point to be made here about the power of impeachment, which is that it doesn't have to be actually exercised to fulfill its intended purpose - the mere threat of it (when made credibly) is the only thing that has ever removed a president from office (so far, at least).