Because the Republicans knew that New York would be decisive in the 1800 elections, they decided that Burr (a New Yorker) should serve as the Virginian Jefferson’s running mate. According to Gordon Wood, no Republican expected Burr to get the same number of electoral votes as Jefferson (p. 282). In fact, one elector was supposed to abstain from voting for Burr, but somehow the Republican's plans got confused--not surprising, as this was still the early days of national party politics. Burr and Jefferson ended up accidentally getting the same number of votes.
How would this be resolved? Most Federalists wanted to throw the election to Burr. Policy-wise, Burr could have been a Federalist as easily as a Republican. He was much more friendly to banks than Jefferson, for one. To Burr, politics was "fun and honor & profit" (Wood, p. 280). Burr was not averse to patronage, which a Revolutionary like Jefferson would label "corruption." He was no ideologue, but merely someone who saw politics as an activity befitting someone of his pedigree and talent (p. 280).
It was Burr's lack of ideological fervor--his overweening concern for himself--that led Federalists to try to throw the election to Burr. Better Burr (whose own self-interest would lead him to play ball with existing Federalist power structures) than the ideologue Jefferson.
Quoting Wood again (284):
Federalists thought they might be able to convince some congressmen to
throw the election to Burr. Indeed, so great was the Federalists' fear
of Jefferson that many of them thought that simply electing Burr was
the best way of keeping Jefferson out of the presidency. Burr, said
the Federalist Theodore Sedgwick of Massachusetts, was a much safer
choice than Jefferson. Burr was no no democrat, he was not attached to
any foreign nation, and he was not an enthusisast for any sort of
theory. He was just an ordinary selfish, interested politician who
would promote whatever would benefit him.
Hamilton, although still friendly with Burr as of 1800, did not agree with this assessment. Hamilton remarked that Jefferson "at least had pretensions to character." Burr, on the other hand, "is sanguine enough to hope every thing--daring enough to attempt every thing--wicked enough to scruple nothing." Even though Hamilton knew "if there be a man in the world I ought to hate, it is Jefferson," he thought the country was safer in the hands of a competent and virtuous (if misguided) ideologue like Jefferson than in a reckless self-server like Burr. Accordingly, Hamilton launched a vigorous campaign for Jefferson (p. 284).
But Hamilton's arguments weren't enough, so deeply did most Federalists distrust Jefferson. Congress voted 35 times without resolving the presidential election. Finally, a Delaware Federalist got assurances second-hand that Jefferson reportedly "would preserve the Federalist financial program, maintain the navy, and refrain from dismissing subordinate Federalist officeholders except for cause." This was enough assurance, and on the 36th ballot, some Federalists abstained from voting, which was enough to throw the presidency to Jefferson over Burr (p. 285).