In the case of Humphrey's Executor v. United States (295 U.S. 602 (1935)), President Hoover nominated
Humphrey to be a Federal Trade Commission (FTC) Commissioner. He was approved
by the Senate. Later, President Roosevelt fired Humphrey because Humphrey didn't
agree with Roosevelt's New Deal policies. Humphrey died (possibly of a broken
heart), and his executor sued for back salary on the theory that the President
could not fire an FTC Commissioner.
The basic point of this case is that if a government official exercises powers that could be considered legislative or judicial, then the President can only fire that official for cause. Officers who have only executive powers can be fired by the President for any reason (see Myers v. United States (272 U.S. 52 (1926))).