Even before there were official term limits for the presidency, it was common practice for Presidents to step down after eight years, dating back to George Washington's decision to retire after his two terms. It wasn't until the 1940s, after President Franklin Delano Roosevelt was elected to a third and fourth term in office, that the idea of amending the Constitution to include a term limit gathered enough steam to be passed. On March 21, 1947, Congress passed the Twenty-Second Amendment - limiting presidents to two terms in office.
"No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this Article shall not apply to any person holding the office of President when this Article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this Article becomes operative from holding the office of President or acting as President during the remainder of such term.
This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several States within seven years from the date of its submission to the States by the Congress."
United States Library of Congress, The Constitution of the United States of America: Analysis and Interpretation
The Twenty-Second Amendment has yet to be tested or applied. Commentary suggests, however, that a number of issues could be raised as to the Amendment's meaning and application, especially in relation to the Twelfth Amendment. By its terms, the Twenty-Second Amendment bars only the election of two-term Presidents, and this prohibition would not prevent someone who had twice been elected President from succeeding to the office after having been elected or appointed Vice President. Broader language providing that no such person "shall be chosen or serve as President . . . or be eligible to hold the office" was rejected in favor of the Amendment's ban merely on election.1
Whether a two-term President could be elected or appointed Vice President depends upon the meaning of the Twelfth Amendment, which provides that no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President. Is someone prohibited by the Twenty-Second Amendment from being elected to the office of President thereby constitutionally ineligible to the office? Note also that neither Amendment addresses the eligibility of a former two-term President to serve as Speaker of the House or as one of the other officers who could serve as President through the operation of the Succession Act.2