The most obvious answer is the election of 1800, but the election on 1796 had some impact on the 12th Amendment as well . The way the constitution was originally structured, electors from each state had 2 votes each with no distinction for president and vice president. It was also the case that many states decided to allot proportional votes (Maryland, Pennsylvania, Virginia, and North Carolina).

The 1796 election between Thomas Jefferson & John Adams was the first contested election as it was the 1st that Washington wasn't running in. The election was pretty nasty even by today's standards with party papers for the Federalists accusing Jefferson of being an anarchist and atheist and Republican papers charging John Adams with having monarchist sentiments and eyeing the creation of a "throne" which his son could inherit. In the end Adams won the electoral college vote with 71 votes to Jefferson's 68. Federalist electors didn't coordinate their "second" votes and split their votes regionally so that Thomas Pinckney (southerner) received 59 votes and Oliver Ellsworth (northerner) received 11. If the Federalists had voted in a block, they would've sent 2 Federalists to the White House. Instead the mix-up allowed Jefferson to finish second and sent him to D.C. to work in the cabinet of a president he had just accused of wanting to destroy the Constitution (Imaging HRC as Trump's VP, it wouldn't be pretty). In the end, Jefferson was relegated to watching over the Senate in what he described as pretty monotonous task (he did oversee a rather unimpressive Andrew Jackson fill an interim Senate term). Jefferson spent his term under the radar and wasn’t serving any advisory roles in the Adam's cabinet. He also played a role in drafting the KY & VA resolutions which sought to directly undermine the Adams supported Alien & Sedition Acts.

In 1800 the same cast of characters ran for the presidency, but this time 2 things had changed: Electors became more disciplined in voting the party preference for President and Veep, and 2 states (NY & VA) independently decided to no longer give proportional electoral college votes. With Aaron Burr's open campaigning (which many saw as unseemly) the Republicans won New York and won the election. Republican electors were too disciplined though and Jefferson and Burr tied with 73 votes each (The Federalist had actually coordinated their votes and allotted Pinckney one less vote than Adams). Burr, being the scoundrel that he was, didn't concede the presidency to Jefferson and, as per the Constitution, the House had to decide the election. This led to 35 separate votes in which Jefferson and Burr kept tying. Eventually on the 36th vote Jefferson was selected as Pres and Burr and VP. Hamilton convinced some Federalists that Jefferson was the least bad option for president-he would undo Federalist policy, but he was at least a known quantity unlike Burr who just seemed power hungry. The whole Hamilton-Burr conflict escalated pretty quickly after that.

So after 2 contested (I mean in the sense that Washington wasn't the unanimous choice) elections it was pretty obvious that no one foresaw how nasty party politics was going to get in 1787 when the Constitution was drafted. So, the 12th amendment stipulated that electors still had 2 votes, but they needed to be marked for President and then Vice-President. The way the text reads it may seem like the intent was to have a separate election for Veep, but it was really about allowing electors to vote for the party ticket.