Not far from the Church on the Mount of the Beatitudes is the Church of the Primacy of St. Peter. The Roman Catholic tradition interprets Jesus comment to Peter in Matthew 16:18, “And I tell you, you are Peter, and on this rock 
I will build my church…” to be appointing Peter as the first Bishop of the Center of the faith in Rome. Being the most important Church, the Bishop of Rome was given deference, if not full allegiance, throughout church history. Thus, the Bishop of Rome has been recognized as the Pope, or leader, of the Catholic (universal) Church. Most Protestants believe the church is built upon Peter’s profession of faith in Jesus as the Messiah rather than the authoritative person. Although the protestant and catholic traditions concerning the significance and primary role of the Apostle Peter differ in several ways, all agree that there was some “primacy” to the ministry of Peter.
You begin to see Peter’s leadership role immediately after our Lord’s ascension. He arranges for the election of St Matthias as the replacement for Judas. He is the spokesman at Pentecost and afterwards, the chief teacher. He works the first miracle, and remains the principal thaumaturge (bringer of miracles). He wields the authority in the case of Ananias and Sapphira. In the early persecutions by the Sanhedrin, he is the ringleader, and he is the one Herod must arrest. When he is in prison, the special prayer of the Church shows the esteem in which he is held.
The Pulpit Commentary lists several more instances that point to Peter’s primacy. 1) Peter was the first-mentioned of the three who saw the transfigured Son of man upon the holy mount. In the garden of Gethsemane, Simon was one of the same band of three whom Jesus kept near to himself. It was his exclamation, “Thou art the Christ, the Son of the living God,” which called forth the Lord’s approval and original blessing, “Blessed art thou, Simon,” etc. Paul himself records that the risen Redeemer first appeared to Peter. As the chief speaker and representative of the Church, he proclaimed, not only the facts of the Resurrection and the outpouring of the Spirit, but pardon and salvation through the redemption which Christ had wrought. Although Paul is recognized as the Apostle to the gentiles, it was Peter who first welcomed the gentiles into the church.