The Vice-President is named successor by the 25th Amendment to the Constitution. After the V-P, Congress fixes the order of succession. The present law which governs the order is the Presidential Succession Act of 1947. To bring it close to home, here is the mandated order if the unthinkable were to happen: 
 

If the Vice-President is sworn in as President, the vice presidency would be vacant. The new President would then nominate a new Vice President, who would take office upon confirmation by a majority vote in both the House and Senate. 

Prior to 1967, when the Twenty-fifth Amendment was ratified, there was no procedure in place for filling a vacancy in the vice-presidency. The office was left vacant until the next election. That happened 16 times in U.S. history. The last instance was when Vice President Lyndon Johnson became President after John Kennedy's assassination in 1963. If the statutory successor does not meet the Constitutional qualifications to President, that individual is passed over.

Since its ratification in 1967, the Twenty-fifth Amendment process has been invoked twice: in 1973 President Richard Nixon nominated Gerald Ford to replace Spiro Agnew, who had resigned mid-term; and in 1974, when Gerald Ford became President after Richard Nixon's resignation. He named Nelson Rockefeller to be Vice-President. Gerald Ford thus holds the distinction in American history of serving as both Vice-President and President without having been elected to either office.