Preface: I am not a historian by any means so this may have an obvious answer but I really want to know.
I was reminded of this fact while listening to the soundtrack to Hamilton and it got me thinking. In modern elections in the USA, the two front runners consist of one primary from each of the two major parties. The primary candidate chooses their running mate (who will be vice president). Then the electoral college is responsible for voting for the team.
Originally, the electoral college decided on the president, and the candidate who got the SECOND most votes was vice president.
It seems to me that this would do away with a lot of the fighting along partisan lines that's caused by a one party administration because usually the candidate who gets the second most votes is of the opposite party.
What caused us to stop using this system?
The Constitution, as you may be aware, contains nothing about parties or party affiliation. The framers were essentially under the belief that educated, patriotic men could always come together and compromise for the benefit of the nation. They could not foresee the division that would take hold even as the Constitution was still drying. Many founders rejected it outright, other wanted a Bill of Rights before they would sign, and others then actively campaigned against it (see Anti-Federalists). So by Washington's presidency, the sides were already forming. The Federalists, led by Hamilton and John Jay, but included Adams and in essence Washington, against the Democratic-Republican (also called Republicans, or Jeffersonian-Republicans) led by Thomas Jefferson and James Madison.
By the end of Washington's term, he was already familiar with the split, which is why in his famous Farewell Address he addressed the partisan divide and urged Americans to avoid political parties. This was too little, too late, by this point. In the 1796 election, Adams was elected, with second place going to his former friend-turned-political-rival Thomas Jefferson. During Adams' presidency, Jefferson on multiple occasions disagreed and even conspired to undermine Adams and his policies, particularly with regards to interaction with France and England.
By 1800, the Federalists lost big. Adams was defeated, and the presidency and both houses of Congress went against the Federalists. The second problem arose in that the Constitution didn't specify who you chose as president vs. vice president, so when Jefferson tied with Aaron Burr for the Democratic-Republican ticket, the vote went to the House to decide. After more than 35 ballots, the vote finally shifted to Jefferson. These two elections were the catalyst for the 12th Amendment, ratified in 1804, that the electors must specify who they choose for president and who for vice-president.
It should be noted, also, that even within this system, some states do not require electors to vote for the candidate who wins the majority of the votes in that state. If they choose to vote against that person, they are called faithless electors. In the most recent 2016 presidential election, there were 8 such votes. For example, Colin Powell received three electoral votes for president, and Elizabeth Warren received two votes for Vice-President.