Here's a useful rundown of the factional splits among the cardinals who are about to undertake "one of the more momentous papal elections." The broad split obviously involves conservative vs. reforming Church factions, but there are other elements that are expected to influence the cardinals' votes.
The relationship with Islam and the challenge of European secularism may make the national origin of the next pope important, and that in turn may be bad news for Italy's cardinals. Even if an Italian is chosen this time, the old Italian lock on the Vatican may be coming to an end. The linked piece argues that "there can be no going back" from John Paul II's globalization of the role.
Of course, these sorts of factional splits are small change compared to the days when the Papacy was an integral part of Europe's power politics. For centuries, such conclaves involved struggles between the monarchies of France and Spain, the Holy Roman Empire, various Italian princes, as well as powerful families of Rome itself.
Under those circumstances, hopeless deadlocks could develop, and conclaves might continue for months or even years before settling on some compromise candidate. (The longest such deadlock lasted nearly three years, from 1268 to 1271.) Rome's citizens could become insistent and even threatening toward deadlocked conclaves: They might wall in the cardinals, greatly reduce their food, or even tear the roof off the building they were using to encourage them to make a choice. (Or rather, to encourage the cardinals' openness to the Holy Spirit, since doctrine held that the choice of a pope was inspired by the Holy Spirit.)
Just such a hopeless deadlock was to result in the most dysfunctional of all Papacies. In 1294, after some 18 months of intrigue and impasse, a conclave suddenly elected an elderly, mountain-dwelling hermit called Peter of Morone. Peter was a popular mystic who was believed to hang his cloak on a sunbeam; when he prayed, an invisible bell was said to toll. The delegation sent to inform him of his election had to scale a sheer, thousand-foot cliff in the Abruzzi to find him. Peter at first refused the office, but was persuaded that he was the expected "angelic pope" who would return apostolic simplicity to Rome, and who would usher in an age of Christian love. It didn't work out that way.
Taking the name Celestine V, Peter never made it to Rome at all. The king of France took immediate control of him, and housed him under his thumb in a Neapolitan palace. Pope Celestine understood nothing of what was going on around him, signed everything laid before him (including blank Papal Bulls), raised to cardinal every French candidate, and grew increasingly miserable. He spent most of the day in prayer.
Soon, Peter began hearing a supernatural voice at night admonishing him for accepting the Papacy at all. That voice is usually attributed to Cardinal Benedetto Gaetani, who entertained papal ambitions of his own, and who supposedly outfitted Peter's room with a speaking tube through which he whispered the heavenly admonitions. By day, the lawyerly Gaetani offered Peter help in extricating himself from the burdensome office. Before the year was out, Celestine had abdicated.
Many common people felt betrayed by the holy hermit in whom they'd placed their hopes for the world's salvation, but it was Peter who was in danger. Benedetto Gaetani quickly engineered his own election to the Throne of Peter, and seeing a living former pope as a potentially serious problem, sought his arrest. Celestine V escaped into the wilderness, and eventually boarded a ship for Dalmatia. The ship was blown back to shore, however, and Celestine was soon locked away in a papal fortress. He's said to have smiled on first seeing his tiny cell, because it reminded him of the huts in which he'd spent his pre-papal life doing penance. He died in that cell within two years.
In 1313, Pope Celestine V was declared a saint. He has an impressive marble tomb in Aquila.