There are two names for the Election of 1800 - the Revolution of 1800... And the Scandal of 1800. It was remarkable for two things. Firstly it was the first peaceful transfer of power between two parties (hence the Revolution) as the Federalists lost over a wave of discontent regarding the Alien and Sedition Acts. The second part (the Scandal) involves two of America's most (in)famous men.
"I hold no illusions that the two are friends; they are rather two snakes who believe the other is fairly in their grasp." - Adams/Jefferson letters (dated July 1805).
Aaron Burr and Alexander Hamilton had indeed been friends during the Revolution as is well known. However by the time of the Election of 1800 they were bitter enemies - worse even than Jefferson and Adams. Burr had defeated Hamilton's father-in-law Phillip Schuyler for a position in New York.
Hamilton and Burr were very similar, both men controlled vast political machines in New York and both were men who intensely craved power. Hamilton and Burr had served together in the army and thus had a bit of a history together. Hamilton had been very influential in the Washington administration and in the Election of 1796 had tried to get Pickney elected over Adams as he thought that the inexperienced Pickney would be more amenable to "advice" from Hamilton. When this didn't work Hamilton was virtually shut out of the Adams administration, frustrating him greatly.
In 1800 Hamilton tried a different tactic, striking a deal with his sworn enemy Aaron Burr. Rather than looking for an ignorant sop to play, Hamilton instead looked to a fellow player to make a deal. When Aaron Burr was elected by the (Federalist) House of Representatives over Thomas Jefferson, many declared it a "Corrupt Bargain". Thanks to getting second place (again) Thomas Jefferson was now Vice President again despite actually winning the election this time. He was understandably frustrated and refused the position, never technically getting inaugurated under Burr. The second-in-line for the Vice Presidency was the Federalist president pro temporare of the Senate James Hillhouse who also declined, leading to the appointment of Massachussetts Democratic-Republican Samuel Adams - who was also the relative of the just defeated former President John Adams.
As a reward for his "help" in winning the election, Hamilton was offered his old role as Secretary of the Treasury in the Burr administration. Ever the social climber, Hamilton adamantly refused and demanded a more high profile position. The seat was thus allowed to continue under Wolcott whom had served in the Washington and Adams administration. Hamilton was then promoted to Secretary of State which had previously been held by Hamilton's own Washington adminstration rival - Thomas Jefferson. The move was seen as a further slap in the face to Jefferson whom decried the whole election as a fraud in private.
Thus Aaron Burr became President under very dubious circumstances (the Scandal of 1800).