The Early Years
The vice presidency was given little consideration by the framers of the Constitution. Some of them were opposed to having the position at all, so as a compromise it was made into a weak office, made possible by a last-minute insertion into the Constitution and given just one explicit duty — to preside over the Senate.
Mess of the 1800 Election
The country's first constitutional crisis occurred during the election of 1800, when Thomas Jefferson and Aaron Burr received the same number of presidential votes in the Electoral College. What made it particularly troublesome was that Burr was Jefferson's running mate!
How did a presidential and vice presidential candidate tie for the nation's top office? Well, the Constitution set up a system in which there was no separate vote for president and vice president. Members of the Electoral College were allowed to submit two votes, but they could not indicate which vote was for president and which was for vice president. The candidate receiving the most votes was elected president, and the runner-up was made vice president. The framers created this odd system because they wanted the vice president to be the second most qualified person in the country (essentially the loser of the presidential election), and in the first two elections it worked.
As the presidential election was being contested in the House of Representatives, Alexander Hamilton worked behind the scenes to deny Aaron Burr a victory. Hamilton detested Burr, and thought he was unfit for the office. Burr blamed Hamilton for his defeat, and four years later (while serving as vice president) challenged Hamilton to a duel, during which he shot and killed him.
What the system didn't anticipate were political parties and running mates. In the election of 1800, Jefferson and his running mate Burr both received 73 electoral votes, while Jefferson's opponent John Adams received only 65. By constitutional mandate, the election was then thrown to the House of Representatives, where on the thirty-sixth ballot Jefferson finally edged out his vice presidential running mate.
As a result of the mess of 1800, the Twelfth Amendment was passed, which called for separate ballots for the election of the president and vice president. Thus, instead of the second most qualified person obtaining the office of vice president (the presidential loser), the president's running mate would become the vice president. A few years later, Congress tried to abolish the vice presidency altogether, but the measure was narrowly defeated in the Senate.
The Veeps History Forgot
The Twelfth Amendment ushered in an era of forgettable — and sometimes regrettable — vice presidents. Few men of national stature were interested in the powerless office, and as a result a rash of vastly underqualified candidates filled the void. Garret Hobart was a New Jersey state legislator when he was tapped as William McKinley's vice president. Chester Arthur, who became president after James Garfield was assassinated, had been a customs collector for the port of New York prior to becoming vice president.
For the seven-year stretch from 1850 to 1857 there was effectively no vice president of the United States. Millard Fillmore became president following the death of Zachary Taylor in 1850, leaving the vice presidency vacant for the remainder of the term. In 1853, newly elected vice president William Rufus King succumbed to tuberculosis only a month into his term, leaving the office unoccupied for the remainder of Franklin Pierce's presidency.
Some nineteenth-century vice presidents had interesting ways of passing the time while in office. Richard Johnson spent much of the Martin Van Buren administration tending bar at his saloon. Ulysses S. Grant's second vice president, Henry Wilson, used the time to pen a three-volume history of slavery. William Rufus King decided to do nothing at all, skipping Franklin Pierce's inaugural to receive medical treatment for tuberculosis in Cuba. He died a month later in his native Alabama without ever serving a day in the office.

