Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
President-elect of the United States
The words "president elect" appear four times in the Constitution, and they didn't appear until 1933, when the Twentieth Amendment, which contained a provision addressing the unavailability of the president elect to take the oath of office on Inauguration Day, was ratified. Section 3 provides that if there is no president-elect on January 20, or the president-elect "fails to qualify", the vice president-elect would become acting president on January 20 until there is a qualified president. The section also provides that if the president-elect dies before noon on January 20, the vice president-elect becomes president. In cases where there is no president-elect or vice president-elect, the amendment also gives the Congress the authority to declare an acting president until such time as there is a president or vice president. At this point the Presidential Succession Act of 1947 would apply, with the office of the Presidency going to the speaker of the House of Representatives, followed by the president pro tempore of the Senate and various Cabinet officers.