Probably the most common word in U.S. v. Nixon after "the," "privilege" refers to the protection of confidential communication from being used in a legal proceeding. Lots of professional relationships have privileged communications. For example, discussions between people and their lawyers, doctors, and clergy are presumed to be completely privileged. There are exceptions, but if information is requested in a subpoena, the professional can attempt to quash it on the grounds of the privilege.

In his motion to quash the subpoena of his Oval Office tapes, Nixon claims the executive privilege: the president's constitutional right to confidential communications free from court inspection. Otherwise, how can he do his job? The executive privilege isn't explicitly mentioned anywhere in the Constitution, but it's assumed from Article II, which delineates the role of the executive as extremely unique.

The Court bends over backwards and does headstands to convey that they totally respect the president's need for confidentiality and that the office of the presidency requires tons of special consideration, etc., etc. Having said that, they decide that because the release of the tapes wouldn't endanger national security or reveal military or diplomatic secrets, the privilege in this case didn't override the prosecutor's need to have the information.

Paradoxically, the Court in this case definitively established the executive privilege as a thing; it had been a murky concept in other cases. So while curbing Nixon's power in this specific instance, the Court expanded presidential power in general.