Was Abraham Lincoln and Jefferson Davis both candidates in the presidential election of 1860?

No, Davis was not a candidate. Davis was a Senator from Mississippi, and did not run for president in 1860. In fact, he had not sought the job as Confederate president. He had resigned from the Senate and was at home, when a telegram arrived informing him he had been chosen as "provisional" Confederate president, by the new Confederate Congress, then meeting in Montgomery, Alabama. Davis had hoped to be named general-in-chief of the Confederate Armies. The new Confederate Congress had written a Confederate Constitution, and then selected Davis as Chief Executive, to serve until regular elections could be held one year later, in February 1862. At that time Davis ran, unopposed, and was elected to the single, six-year term provided for by the Confederate Constitution.

Lincoln had three opponents in the 1860 election. The Democrats split into northern and southern factions. The northern faction nominated Stephen Douglas, who had beaten Lincoln in the 1858 Illinois Senate race - the only election Lincoln ever lost. The southerners walked out of the Democratic Convention, held their own convention, and nominated John C. Breckenridge, who was Vice President to Buchanan at the time. A third party formed, the Constitutional Union Party, and nominated John Bell.

With those three dividing the opposition vote, Lincoln won the 1860 election with about 40% of the popular vote, and carried no southern states at all.