In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
Before that it was unclear if the vice president of the United States became the president or an acting president. After the first death of a president in office, William Henry Harrison, on April 4, 1841, John Tyler stepped up and took the oath of office. The Tyler Precedent was followed by Millard Fillmore, Andrew Johnson, Chester Arthur, Theodore Roosevelt, Calvin Coolidge, Harry Truman, and Lyndon Johnson.
The amendment's second section clarified what to do if there is a vacancy in the vice presidency. Whenever a vice president became president, like those listed above, or died in office or resigned, the vice presidency remained unoccupied. Six years after the passage of the amendment, Spiro Agnew became the second vice president to resign from office and Richard Nixon used the new amendment to fill the office. Gerald Ford became the first vice president appointed to the office, after approval by both Houses of Congress. Less than a year into the office, Nixon became the first president to resign the presidency making Ford the first president of the United States never elected to either office. Once president, Ford appointed Nelson Rockefeller to the vacant office of the vice presidency. So from 1974-1977, the POTUS and VPOTUS were appointed office holders.
The third section of the amendment also allows the president to transfer power to the vice president, if the president is unable to perform the duties of the office. President Ronald Reagan did so with Vice President George Bush and President George W. Bush did it twice with Vice President Richard Cheney. Previously this should've happened with Presidents James Garfield, Grover Cleveland, William McKinley, Woodrow Wilson, and Dwight D. Eisenhower, but there was no amendment then, of course. To transfer power, the president must write a declaration of of the transfer to the President Pro Tempore of the Senate and the Speaker of the House of Representatives, both relinquishing the powers of the office to the vice president and then resuming the duties of the office.
The final section of the amendment has never been put in to effect. President George H.W. Bush calls it a legal coup since it requires the vice president and a majority of the cabinet to declare the president unfit for office. The movie Air Force One shows the vice president and cabinet with section 4 of the amendment as a backup plan while the president is held hostage by terrorists, but the plan is never used.