I'm sorry if my question sounds stupid but I'm not an American citizen, is it the first time the US ends up with a such long delay to proclaim their new president?
1800 was the longest, the Presidency wasn't decided until 17th February 1801.
Nowadays each elector casts a vote for President and another for Vice President, but back in 1800 they cast two votes with the candidate getting the most votes being the President and the runner up becoming the Vice President. This worked in 1788 and 1792, everyone voted for Washington but had different views on who should be VP, in 1796 it messed up, the Adams/Pinckney ticket should have won but not enough electors voted for Pinckney meaning that Jefferson (the defeated Presidential candidate) ended up second and Vice President even though he was from a different party.
Both parties were more organised for 1800, this time everyone would vote for the head of the ticket except for one who would cast their second vote for someone else. So for the Federalists, Adams got 65 votes, his VP candidate Pinckney got 64 and Jay got 1. But something went wrong on the Democratic-Republican side of the election, both Jefferson and Burr got 73 votes, a tie for the Presidency. The Constitution to the rescue, a contingent election in the House, each state's delegation to cast a vote for either Jefferson or Burr. All the Democratic-Republicans voted for Jefferson, the Federalists didn't really like Burr, but at least he wasn't Jefferson. There were 16 states at the time, 8 states voted for Jefferson, 6 for Burr and 2 (Maryland and Vermont) were divided. No majority for either candidate so they had a second ballot, same result, a third, a fourth and so on. It was a week later on the 36th time of asking that some of the Federalists decided to not cast their ballots, 2 states flipped from deadlock to Jefferson, 2 states flipped from Burr to deadlock and Jefferson was elected President 10-4 with Burr as his Vice President.
Jefferson went on to serve two terms, the second with George Clinton as his Vice-President. Burr went off to shoot Hamilton in the infamous duel and get tried for treason. The US passed the Twelfth Amendment to bring the voting system firmly into the 19th century so we didn't have this flavour of mess again.