After months of discussion and debate, the cardinals obedient to Avignon and those obedient to Rome joined in Conclave. Canon Law declared that an election not take place until 10 days after the death of a pope, but the cardinals agreed that the Papal See had been vacant for at least that long. The Conclave of cardinals went on for 11 days, while the general Council of Pisa continued separately. The Council felt that, if the cardinals could not find a suitable candidate who was agreeable to all, they would intervene and have their own election. This was not allowed by Canon Law, however, and did not get much traction.
Finally, Baldassare Cardinal Cossa convinced the Conclave to vote unanimously for Peter of Candia. He had been made a cardinal by Innocent VII in 1405, but had worked to heal the division between popes. On 26 June he was elected Pope Alexander V. He presided over the remaining council meetings, pledged to work for reform, and confirmed appointments and ordinations made by the various cardinals.
The two (now) antipopes repudiated the outcome of the Council of Pisa, and an interesting dilemma was raised: if Popes Benedict and Gregory were deemed illegitimate, then did that mean that their appointment of cardinals was illegitimate, and if so, could these cardinals have the authority to elect a pope? The Council of Pisa was nor received by everyone with a sigh of relief that the schism was over; there were serious concerns about its ability to elect Alexander.
In act, the controversy continued into the 20th century. Let me explain.
Alexander V's time as pope was brief: he died 3 May 1410, having been pope less than a year (that is his tomb above). He was succeeded by none other than his strongest proponent at Pisa, and Cardinal Cossa became Pope John XXIII. Keep in mind that Benedict in Avignon and Gregory in Italy never "resigned" their titles, so there were still three men claiming to be pope. In fact, Benedict outlasted John and Gregory.
The Annuario Pontificio ("The Pontifical Yearbook") listed the Roman popes as legitimate until 1409, followed by the Pisan popes elected by the reconciled cardinals; that means Gregory XII, Alexander V, and John XXIII were "official" (even though Gregory and Alexander were claiming legitimacy concurrently). I say "were" because that changed in 1958 when Angelo Cardinal Roncalli was elected pope to replace Pius XII. He chose the papal name John XXIII, claiming there had been only 22 legitimate Pope Johns. The Annuario Pontificio was re-written to indicate that the Roman Pope Gregory's time in office was officially recognized as lasting until 1415, to be succeeded by Pope Martin V.
So all that work to find a compromise candidate in Peter of Candia to become Alexander V and heal the schism has been quietly ignore, and "Alexander V" and the first "John XXIII" are now deemed antipopes.
[edit] Of course, nothing required a pope to be in Rome, and Avignon wasn't the only non-Rome location for some popes. And speaking of Popes: who was the first pope? I know what you are thinking, but wait until tomorrow for the real answer.