The Great Schism discussed quite well above by mwestwood is the most famous schism, although well before the Reformation, but there have actually been other points in time at which there have been rival popes. When Urban II gave his speech at Clermont, France in 1095 calling for a Crusade to free the Holy Land from the Turks, he had three motives. One was the persecutions the Turks were visiting upon Christians (and Jews and Arabs), another was the underlying inflationary economy of Europe caused by trade with the Middle East. But possibly the most important was that he had a rival pope, Clement III, who had great support in the Holy Roman Empire. In the enthusiasm for the Crusade, Clement III's support dwindled to zero.
Curiously enough, there is still a rival pope. In 1978, on the death of Pope Paul VI, a blind Spaniard named Clemente Dominguez y Gomez claimed he had a personal revelation from the Virgin Mary and proclaimed himself Pope Gregory XXVII, and true head of the Catholic Church. Dominguez claimed the Roman Catholic Church was in the clutches of Satan, and he was chosen to free it. He eventually amassed about 150 clerics, including priests, nuns, bishops and cardinals, and over 1,500 lay members around the world as the Holy Dogmatic Palmyrian Synod. On his death in 2005 he was succeeded by Peter II.
During the Reformation.
In the fourteenth century the Catholic Church had two popes. The Papal election of Urban VI in 1378, the year after the papacy returned to Rome, Italy, from Avignon, France, greatly displeased the French cardinal who were used to the luxurious life that Urban, an austere man, scorned.
Thirteen cardinals withdrew to seek a more agreeable successor and elected Robert of Geneva as Pope Clement VII. Clement returned to Avignon. So there were two popes, one in Rome, Italy, and one in France. In 1394 the University of Paris declared that itself in favor of an abdication by both popes. Clement died that year, but his successor Benedict XIII was determined to cling to his office.
In 1415 there were three popes. After the council of 1414 arguments continued, but, finally in 1515, two of the popes resigned.