In the Gospel, Jesus called Peter the rock on which he would build his church, and he went on to say, "Whatever you bind on earth shall be bound in heaven.
However, Jesus was not saying that our popes stand alone as God's voice on earth. Two chapters on in Matthew's Gospel we read where Jesus, speaking to his other disciples said, "Whatever you bind on earth will be bound in heaven."
After the year 100, Europe was overrun by pagan peoples from the East. There were the Goths, the Visigoth, the Francs, the Vandals; and all of them saw their kings as possessing unlimited authority directly from heaven. Then, as those nations adopted Christianity, the popes too were being seen as possessing all authority directly from heaven.
But, Jesus was also speaking to Peter when he said, "You know how among the Gentiles those who exercise authority lord it over them but it cannot be that way with you."
St. Cyprian in 255 had two things to say about the preeminence of the popes. On one hand, he said that Christian communities everywhere, if they were to be seen as belonging to the Catholic Church, must stay united to Rome and to the popes. On the other hand, St' Cyprian denied the pope's right to appoint bishops to African dioceses.