Under the U.S. Constitution as originally ratified, when I presidential election was held, the candidate who received the most vote became president, and the candidate who received the second-most votes became vice president.
After a couple of administrations -- most notably John Adams's presidency from 1797-1801, where Thomas Jefferson, a political opponent, served as his vice president -- Congress quickly realized that this provision was unworkable. Congress passed and the states ratified the 12th Amendment in June 1804, which created the system still followed today, where the president and vice president run together as a ticket, and voters voted for tickets.
That was 220 years ago. In the ensuing years, and spanning 55 presidential elections, a sitting vice president has been elected president only twice: Martin Van Buren in 1836, and George H.W. Bush in 1988. That's it.
In fact, there have more sitting VPs who lost their election bids than those who won them: Al Gore (2000), Hubert Humphrey (1968), Richard Nixon (1960), and John C. Breckinridge (1860).