Popes are elected for life unless they voluntarily — without pressure or coercion — resign from office. Pope Benedict XVI resigned early in 2013, claiming fading bodily and mental strength; he was the first pope to resign since 1415.
Pope Pontian was the first one to abdicate from the office in A.D. 235. Pope St. Peter Celestine V was the most famous one to resign, going back to monastic life in 1294. Pope Gregory XII quit in 1415.
No one can depose a pope even if he becomes insane, sick, or corrupt. No ecumenical council has the authority to remove him from office. So when a bad pope gets in (and from time to time, a bad pope has been elected), the only course of action is to pray to St. Joseph for a happy death of the pope in question.
St. Joseph is the patron of a happy death, because he probably died of natural causes in the arms of Mary and Jesus.
Although even one bad pope is one too many, Jesus himself picked 12 imperfect sinners to be his apostles. The first pope, St. Peter, weakened and denied Christ three times, and Judas, one of the first bishops, betrayed him for 30 pieces of silver. One repented; the other hanged himself instead of seeking mercy.
Of the 265 popes in history, only a dozen were real scoundrels and caused great scandal. Seventy-eight popes are recognized as holy saints, leaving 175 pretty good, all right guys. Better stats than for presidents, prime ministers, or monarchs around the world.