During the third century, the Emperor Decius sought to eliminate Christianity by exterminating its adherents. He wasn't the first or the last to try this, but he was among the most determined. In fact, he had done such a number on the faithful that after Pope Fabian got killed, no one stepped up to sit on Peter's chair for the next fourteen months. The heir apparent, a man named Moses, got killed before they could elect him. Those who weren't killed went to ground and stayed there until the killing subsided.
The Goths were raising a ruckus up north, forcing Decius to take his eyes off the cross. Once the Emperor was out of Rome, the Christians popped back up and started quarreling. One leader, Novatian, was adamant that anyone who apostatized during the persecution should not be accepted back into the fold. No bishop could assure them of forgiveness, he argued; they'd have to wait until judgment day to see if that sin could be absolved.
Although Cornelius was not eager to be pope, neither did he want Novatian to get the Chair. After all, he was just as certain that the successor to Peter, who denied Jesus three times and was forgiven, could assure a penitent apostate of forgiveness. Cornelius won the election, but Novatian set himself up as an anti-pope. Cornelius then called a synod to get himself confirmed as pope. Novatian did not accept the ruling of the synod, was excommunicated, and led a schismatic church that lasted a couple of centuries.
The Goths, meanwhile, managed to kill the Emperor Decius in battle. Trebonianus Gallus became the next emperor (there's more to that story but it's not relevant here) and revived the persecutions. Poor Cornelius' papacy last about twenty-seven months, though he spent about a year of it exiled to the coastal Italian town of Centumcellae. He is listed as a martyr. One source said he died of the hardships of exile. Another said he was beheading, which would certainly be considered a hardship, whether or not it happened in exile. His successor, Lucius I, was elected promptly after Cornelius' death, and even though he was not killed in the subsequent persecution, he did not even last a full year.