Papal Primacy

"And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it. And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven."

Introduction

Papal primacy refers to the supreme episcopal jurisdiction of the pope as pastor and governor of the Universal Church. As Catholics, we believe St. Peter was the Prince of the Apostles, the Vicar of Christ, and our first Pope. We also believe that our current Pope is the successor of St. Peter, and is vested with his same authority. The expression "power of the keys" is derived from Christ's words to St. Peter (Mt. 16:19). The promise there made finds its explanation in Isaiah 22:22, in which "the key of the house of David" is conferred upon Eliacim, the son of Helcias, as the symbol of plenary authority in the Kingdom of Juda. Christ by employing this expression clearly designed to signify his intention to confer on St. Peter the supreme authority over His Church.

Catechism
The Lord made Simon alone, whom he named Peter, the "rock" of his Church. He gave him the keys of his Church and instituted him shepherd of the whole flock."The office of binding and loosing which was given to Peter was also assigned to the college of apostles united to its head.” This pastoral office of Peter and the other apostles belongs to the Church's very foundation and is continued by the bishops under the primacy of the Pope.
- Catechism of the Catholic Church, 2nd ed., (1997), n.881
The Pope, Bishop of Rome and Peter's successor, "is the perpetual and visible source and foundation of the unity both of the bishops and of the whole company of the faithful." "For the Roman Pontiff, by reason of his office as Vicar of Christ, and as pastor of the entire Church has full, supreme, and universal power over the whole Church, a power which he can always exercise unhindered."
- Catechism of the Catholic Church, 2nd ed., (1997), n.882
"The college or body of bishops has no authority unless united with the Roman Pontiff, Peter's successor, as its head." As such, this college has "supreme and full authority over the universal Church; but this power cannot be exercised without the agreement of the Roman Pontiff."
- Catechism of the Catholic Church, 2nd ed., (1997), n.883
Scripture

Mt. to Rev. - Peter is mentioned 155 times and all other apostles combined are 130 times. Peter is always listed first but for two obvious exceptions to the rule (1 Cor 3:22; Gal 2:9).

Mt. 16:16; Mk. 8:29; Jn. 6:69 - Peter is first among the apostles to confess the divinity of Christ.

Mt. 16:17 - Peter alone is told he has received divine knowledge by a special revelation from God the Father.

Mt. 16:18 - Jesus builds the Church only on Peter with the other apostles as the foundation and Jesus as the Head.

Mt. 16:19 - Only Peter receives the keys, which represent authority and dynastic succession to his authority.

Jn. 21:15-17 - Jesus tells Peter to "feed my lambs" "tend my sheep" "feed my sheep". Peter feeds all, including apostles.

Mt. 17:23 - Peter is asked for Jesus' tax. Peter spoke for Jesus with authority when Jesus wasn't there. He is the Vicar of Christ.

Gal. 1:18 - Paul spends fifteen days with Peter before beginning his ministry, even after Christ's Revelation to Paul.

Acts. 1:20-26 – By word of Peter, a new bishop/apostle is chosen by lots under the guidance of the Holy Spirit.

Acts. 5:1-6 – Peter passes judgment on Ananias and his wife for their incorrect behavior as disciples.

Acts. 15:7-12 – Peter resolves a doctrinal issue. After Peter spoke, all were silent. Paul and Barnabas speak in support

Tradition & Fathers
"The church of God which sojourns at Rome to the church of God which sojourns at Corinth ... But if any disobey the words spoken by him through us, let them know that they will involve themselves in transgression and in no small danger."
- Pope St. Clement of Rome, 1st Epistle to the Corinthians (A.D. 96)
"… the Church which presides in the place of the region of the Romans, and which is worthy of God, worthy of honour, worthy of the highest happiness, worthy of praise, worthy of credit, worthy of being deemed holy, and which presides over love..."
- St. Ignatius of Antioch, Epistle to the Romans (A.D. 110)
" [we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its pre- eminent authority..."