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The Roman Catholic Church acknowledges1 the direct Biblical basis for Peter's primacy comes from Matthew 16:17-19:
17Jesus said to him in reply, “Blessed are you, Simon son of Jonah. For flesh and blood has not revealed this to you, but my heavenly Father.
18And so I say to you, you are Peter, and upon this rock I will build my church, and the gates of the netherworld shall not prevail against it.
19I will give you the keys to the kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven.”
Additionally, there's this passage from John 21:15-17:
15When they had finished breakfast, Jesus said to Simon Peter, “Simon, son of John, do you love me more than these?” He said to him, “Yes, Lord, you know that I love you.” He said to him, “Feed my lambs.”
16He then said to him a second time, “Simon, son of John, do you love me?” He said to him, “Yes, Lord, you know that I love you.” He said to him, “Tend my sheep.”
17He said to him the third time, “Simon, son of John, do you love me?” Peter was distressed that he had said to him a third time, “Do you love me?” and he said to him, “Lord, you know everything; you know that I love you.” [Jesus] said to him, “Feed my sheep.
And as DJClayworth noted in the comments, his role as the leader of the Apostles (and subsequently, the early Church) after Jesus's death and resurrection is clearly shown, particularly in the first half Acts of the Apostles which almost exclusively features his mission.
Note 1: The Catechism of the Catholic Church states:
552Simon Peter holds the first place in the college of the Twelve; Jesus entrusted a unique mission to him. Through a revelation from the Father, Peter had confessed: "You are the Christ, the Son of the living God." Our Lord then declared to him: "You are Peter, and on this rock I will build my Church, and the gates of Hades will not prevail against it." Christ, the "living Stone", thus assures his Church, built on Peter, of victory over the powers of death. Because of the faith he confessed Peter will remain the unshakeable rock of the Church. His mission will be to keep this faith from every lapse and to strengthen his brothers in it.
553Jesus entrusted a specific authority to Peter: "I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." The "power of the keys" designates authority to govern the house of God, which is the Church. Jesus, the Good Shepherd, confirmed this mandate after his Resurrection: "Feed my sheep." The power to "bind and loose" connotes the authority to absolve sins, to pronounce doctrinal judgements, and to make disciplinary decisions in the Church. Jesus entrusted this authority to the Church through the ministry of the apostles and in particular through the ministry of Peter, the only one to whom he specifically entrusted the keys of the kingdom.
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Mark Trapp had a nice biblical answer, but here's another nugget:
John 20:3-5 (DRA)
3 Peter therefore went out, and that other disciple, and they came to the sepulchre.
4 And they both ran together, and that other disciple did outrun Peter, and came first to the sepulchre.
5 And when he stooped down, he saw the linen cloths lying; but yet he went not in.
6 Then cometh Simon Peter, following him, and went into the sepulchre, and saw the linen cloths lying,
Here, John shows deference to Peter, as the leader he should be the first to inspect the empty tomb. That deference could just be because of age, but it is clear that John thought Peter was the one who should inspect the tomb first. If he didn't he A.) would have went in first and B.) wouldn't have bothered writing about it so many years later.