In the 1796 election, it resulted in John Adams being elected President and Thomas Jefferson being elected Vice President. Political parties were forming up, and the two were on opposing sides and didn't get along very well.
The crap really hit the fan in the 1800 election, when the Democratic-Republican Party ran Thomas Jefferson and Aaron Burr. They intended that Jefferson would be elected President and Burr would be elected Vice President, but it never occurred to them to make sure that one of their electors would vote for Jefferson and not vote for Burr. The result was a tied electoral vote.
The election was then put into the hands of the House of Representatives -- the incumbent one controlled by Federalists, not the incoming one. The House remained deadlocked for some time, as many of the Federalists voted for Burr to throw sand in the enemy gears. Eventually, Alexander Hamilton (who had respect for Jefferson, and none whatsoever for Burr) did enough persuasion and arm-twisting to break the logjam. (This became one of the grievances that eventually led to the duel in which Burr killed Hamilton.)
Thanks! It never fails to amaze me the facts and history my fellow Freepers have at their fingertips!