The Catholic Church has always and will always have one legitimate and authoritative pope. Although there have been times in history when multiple people claimed to be the pope and there were times when conclave took a while, Pope Francis is the most recent direct successor of Saint Peter. Furthermore, another super awesome thing about the Catholic Church and the unbroken line of apostolic succession is that every single catholic priest can trace the bishop who ordained him to the bishop who ordained that bishop and so on all the way to one of the original apostles. If someone wants to see a list of all the popes and when they were pope, that can be done here:
Until the 11th century, Catholicism was a loose religion. Different monarchs (such as the Visigoths and Franks) adopted the Christianized religion, but there was no central base. Rome didn't become the capital of Catholicism until 10something. It's only then that the Papal States took form. So it's a more modern idea in comparison.