The vice presidency of Aaron Burr Jr. was the third vice presidency from 1801 to 1805 during Thomas Jefferson's first presidential term. Aaron Burr is mostly remembered for his personal and political conflict with Jefferson and Alexander Hamilton that culminated in the Burr–Hamilton duel where Burr killed Hamilton, and multiple trials for treason in what became known as the Burr conspiracy.

Burr was born to a prominent family in what was then the Province of New Jersey. After studying theology at Princeton University, he began his career as a lawyer before joining the Continental Army as an officer in the American Revolutionary War in 1775, returning practicing law in New York City, where he became a leading politician and helped form the new Jeffersonian Democratic-Republican Party, then represented New York United States Senate from 1791 to 1797. Burr ran as the Democratic-Republican vice presidential candidate in the 1800 election. An Electoral College tie between Burr and Thomas Jefferson resulted in the U.S. House of Representatives voting in Jefferson's favor, with Burr becoming Jefferson's vice president due to receiving the second-highest share of the votes. The debacle lead to the 12th Amendment to the U.S. Constitution which changed the vice presidency to run with the president rather than being awarded to the runner-up candidate. Although Burr maintained that he supported Jefferson, the president never trusted Burr, believing he sought to become president in 1800 instead.

Jefferson relegated Burr to the sidelines of the administration during his presidency. As it became clear that Jefferson would drop Burr from his ticket in the 1804 presidential election, Burr chose to run for the governorship of New York instead. He was backed by members of the Federalist Party and was under patronage of Tammany Hall in the 1804 New York gubernatorial election. Hamilton campaigned vigorously against Burr, causing him to lose the gubernatorial election to Morgan Lewis, a member longtime New York Governor George Clinton's Democratic-Republican who Hamilton endorsed. Burr then challenged Hamilton to a duel at dawn on July 11, 1804. In the duel, Burr shot Hamilton in the abdomen. Hamilton returned fire and hit a tree branch above and behind Burr's head. Hamilton was transported across the Hudson River for treatment in present-day Greenwich Village in New York City, where he died the following day, on July 12, 1804. This also marked the death of Burr's political career, as he was vilified for shooting Hamilton. Burr was indicted for dueling, but all charges against him were dropped. Burr became the first vice president to be dropped from a presidential ticket when George Clinton was selected as Jefferson's running mate in 1804.

After his vice presidency, Burr traveled west to the American frontier, seeking new economic and political opportunities. Jefferson maintained his distrust of Burr, whose secretive activities led to an 1807 arrest in Alabama on charges of treason. Burr was brought to trial more than once for what became known as the Burr conspiracy, an alleged plot to create an independent country led by Burr, but he was acquitted each time. Burr moved to Europe from 1808 to 1812 before returning to the United States, dying on September 14, 1836, at the age of 80.

In the 1800 United States presidential election, Burr combined the political influence of the Manhattan Company with party campaign innovations to deliver New York's support for Thomas Jefferson. That year, New York's state legislature chose the presidential electors, as they had four years earlier, in 1796, when they gave their support to John Adams. Prior to the April 1800 legislative elections, the State Assembly was controlled by the Federalists. The City of New York elected assembly members on an at-large basis. Burr and Hamilton were the key campaigners for their respective parties. Burr's Democratic-Republican slate of assemblymen was elected, giving the party control of the legislature, which in turn gave New York State's electoral votes to Jefferson and Burr. This drove another wedge between Burr and Hamilton, who had developed a rivalry with Jefferson. Burr enlisted the help of Tammany Hall to win the voting for selection of Electoral College delegates.

Burr was chosen to be the junior member of the Democratic-Republican presidential ticket with Jefferson in the 1800 election, and carried his home state of New York. The Democratic-Republican Party planned to have 72 of their 73 electors vote for Jefferson and Burr, with the remaining elector voting only for Jefferson. The electors failed to execute this plan and Burr and Jefferson were tied with 73 votes each. The Constitution stipulated that if two candidates with an Electoral College majority were tied, the election would be moved to the House of Representatives—which was controlled by the Federalists, at this point, many of whom were loath to vote for Jefferson. Members of the Democratic-Republican Party understood they intended that Jefferson should be president and Burr vice president, but the tied vote required that the final choice be made by the U.S. House of Representatives, with each of the sixteen states having one vote, and nine votes needed for election.

Burr remained quiet publicly, refusing to surrender the presidency to Jefferson, who was seen as the great enemy of the Federalists. Rumors circulated that he and a faction of Federalists were encouraging Democratic-Republican representatives to vote for him, blocking Jefferson's election in the House. However, solid evidence of such a conspiracy was lacking, and historians generally gave Burr the benefit of the doubt. In 2011, however, historian Thomas Baker discovered a previously unknown letter from William P. Van Ness to Edward Livingston, two leading Democratic-Republicans in New York. Van Ness was very close to Burr, serving as his second in the duel with Alexander Hamilton. As a leading Democratic-Republican, Van Ness secretly supported the Federalist plan to elect Burr as president and tried to get Livingston to join. Livingston agreed at first, then reversed himself. Baker argues that Burr probably supported the Van Ness plan: "There is a compelling pattern of circumstantial evidence, much of it newly discovered, that strongly suggests Aaron Burr did exactly that as part of a stealth campaign to compass the presidency for himself." The attempt did not work, however, at least in part because of Livingston's reversal and especially Hamilton's vigorous opposition to Burr.

Although Hamilton had a long-standing rivalry with Jefferson stemming from their tenure as members of George Washington's cabinet, he regarded Burr as far more dangerous and used all his influence to ensure Jefferson's election. On the 36th ballot, the House of Representatives gave Jefferson the presidency, with Burr becoming vice president.