Franklin Pierce became president in the last election, but by the time of this presidential race the slavery issue was tearing the country apart. Pierce was a northern politician with Southern sympathies and thus lost much support in some regions. He lost the Democratic nomination, being the first sitting president to do so.
James Buchanan, Minister to Great Britain nabbed the Democratic nomination under Pierce's feet.
John C. Fremont became history's first Republican candidate. He was a former Senator from California.
Former President Millard Fillmore ran on the 3rd party ticket for the Know Nothing party.
The Republicans came straight out as the anti-slavery party, and strongly campaigned on the platform. Democrats were tolerant of it, and claimed a Republican victory would end in civil war. They were right about that 4 years later.
It was a pretty easy victory for Buchanan with him winning a majority of states. Fremont only won the up-most northern states. Fillmore managed to win one state.