The 25th Amendment to the U.S. Constitution provides the procedures for replacing the president or vice president when they are removed from office. in Section 1, it states that in the even of a presidential vacancy (either for death or resignation), the Vice President shall become President.
When Richard Nixon decided to resign in August 1974, in order to avoid impeachment (he was involved in the Watergate Scandal), Gerald Ford, who was serving as the 40th Vice President at the time, assumed the Presidency, and also became the first President of the United States that was not elected by the United States Electoral College. These actions were the result of having followed the 25th Amendment procedures.