The idea of the top two candidates being named President and Vice President had been devised by the framers in the original Constitution. The belief was that political parties and factions would not appear and that the top two vote receivers in a general election would be a fair way to determine who serves as President and Vice President, respectively. As referred to the in previous post, the animosity between both candidates in the Election of 1800 caused a change in this thought process. With this in mind, the 12th Amendment to the Constitution was developed. This stated that the election of the President and Vice President would be done as a "ticket," whereas a person is able to vote for both president and vice president. What became common practice is for the presidential candidate had the right to select their vice presidential running mate.
Under the original system, however many candidates would run for president. The electoral college would vote on the candidates. Each elector would vote for two people. After the electoral college had done its vote, whichever candidate got the most votes would be the president.
Unlike today, however, the candidates would not have vice presidential candidates already picked. Instead, whoever got the second most votes would become the vice president. This was a bad system because, as it turned out, you could get presidents and vice presidents who were political rivals, as when Aaron Burr became Thomas Jefferson's vice president.