Duels were fairly common at the time and their rivalry went back years. In the 1796 and 1800 elections Burr and Hamilton were bitter political enemies. Hamilton publicly attacked Burr calling it a religious duty to oppose him.
The 1800 election was chaotic. Jefferson and Burr tied with 73 electoral votes each. Under the rules then the House of Representatives had to decide. After 35 ballots Federalists led by Hamilton supported Jefferson as the lesser evil. Jefferson became president and Burr vice president.
By 1804 Jefferson had dropped Burr. Burr ran for governor of New York as an independent but Hamilton campaigned hard against him and attacked his character. Burr challenged Hamilton to the duel to restore his reputation.
Accounts differ on the shots but Burr hit Hamilton in the stomach. Hamilton died the next day. Burr was later charged with treason, acquitted on a technicality and lived the rest of his life in disgrace. Hamilton’s face ended up on the ten dollar bill.
Compared to today’s elections the campaigns of the 1800s were far nastier and more violent, so far. The Constitution handled the 1800 tie and still works today.