Jesus, in telling Peter to feed his lambs and his sheep, gave him a superior position among his apostles; but he did not give any precise description of his superior status. Subsequent events might have altered our understanding of Peter's powers.
For one thing, in 250 a.d. Cyprian, the Bishop of Carthage, called on all of Christendom to see union with Rome as the essential requirement of being called Christian. However, Cyprian denied the pope's right to name bishops outside of Rome. The First Vatican Council seemed to go against that in ceding universal jurisdiction to the popes.