Along-term effect of the election of 1800 was

The election of 1800 ended in a tie between Aaron Burr and Thomas Jefferson. The vote became up to the delegates and ultimately up to Alexander Hamilton. Jefferson eventually won the election becoming president.

A long term effect of this election is that it resulted in the 12th amendment which changed the way we elect presidents today. This was proposed in 1803 and ratified in 1804. This was mainly caused because of Jefferson being the president at the time.

(And just in case you need more background on the 12th amendment) :

The 12th Amendment specifies that the Electors now chose a candidate for President and a separate candidate for VP. Ties for President are decided by the House. Ties for the VP are decided by the Senate.
It is therefore possible to have a President and VP of different parties.