The detail
From 1309 the popes resided at Avignon, and the papacy was widely seen as dependent on the French crown. Gregory XI returned to Rome in 1377 and died there the next year. In April 1378, under pressure from a Roman crowd demanding an Italian pope, the cardinals elected the archbishop of Bari, who took the name Urban VI. His harsh conduct alienated them, and in September 1378 most of the same cardinals, meeting at Fondi, declared the earlier election invalid because it had been made under duress and elected Robert of Geneva as Clement VII, who settled in Avignon.
This was not a dispute over doctrine. Both lines professed the same faith, sacraments and understanding of the papacy; the question was simply which claimant was the lawful pope. Europe split along political lines: France, Scotland, Naples (at times), Castile and Aragon generally followed Avignon; the Empire, England, much of Italy, Hungary and Poland generally followed Rome. Saints and serious theologians stood on both sides, for example Catherine of Siena with Rome and Vincent Ferrer, for a time, with Avignon.
Attempts to resolve the schism by negotiation or abdication failed. In 1409 cardinals from both obediences met at the Council of Pisa, declared both popes deposed and elected Alexander V (succeeded in 1410 by John XXIII). Since neither existing claimant yielded, there were now three. Finally, at the urging of Emperor Sigismund, the Council of Constance (1414–1418) deposed John XXIII (1415), accepted the resignation of the Roman pope Gregory XII (1415), who first formally convoked the council in his own name, and deposed the Avignon pope Benedict XIII (1417). On 11 November 1417 it elected Martin V, who was accepted almost everywhere.
The schism had lasting consequences. It damaged the prestige of the papacy and strengthened conciliarism, the view that a general council holds authority over the pope in certain cases, expressed in Constance's decree Haec sancta (1415) and its plan for regular councils in Frequens (1417). The papacy later rejected the superiority of councils over popes, and how much authority Haec sancta carries is still debated among Catholic historians and canonists. The same Council of Constance condemned John Wyclif and had Jan Hus burned (1415), events connected to later reform movements.
How it is told: the Catholic Church today, in its official list of popes, treats the Roman line (Urban VI to Gregory XII) as legitimate and the Avignon and Pisan claimants as antipopes; that is why a 20th-century pope could take the name John XXIII again. Historians note that contemporaries acting in good faith could not easily tell who was right. Protestant and Orthodox writers have sometimes cited the schism as evidence against papal claims, while Catholic writers stress that the unity of faith was never broken and the crisis was resolved within the Church's own structures.