There were two Democrat tickets, one headed by Vice President John C. Breckinridge, the other by Senator Steven Douglas. And there was the last remnant of the Whigs, called the Constitutional Union Party, led by a man named John Bell.
Lincoln won in that divided field.
Former Tennessee Senator John Bell's Constitutional Union Party was a big deal in Border States and he could actually carry more states & win more electoral votes than Northern Democrat Stephen Douglas.
Bell illustrates why Tennessee and other Upper South states will refuse to secede over slavery or anything else, but then could flip sides if the Union threatens military actions.
Bell's home town was Nashville, Tennessee, just west of the rough dividing line within Tennessee between the Unionist East and Confederate West.
Near as I can tell, after the events of early 1861 Bell played no further role in politics and died in 1869, age 73.