Catholics see this as a foundation for the papacy; in which Peter is named the head apostle and is given special authority (actually a lot of that comes from verse 19 when Peter is given the keys).
To undermine the papacy many people say Peter wasn’t the rock at all. They say Jesus was the rock by bringing up the Greek Petra/Petros or they’ll say Peter’s confession of Christ in verse 16 is “the rock”. Or maybe Peter’s faith is the rock? Who knows, it could be anything; other than the obvious, of course.
These are nice theories and I’m sure a lot of people believe them. What about the early Christians? What did they say about the matter? Did they think Peter was the rock on whom the Church is built or was Jesus saying something else?
“Peter, who is called 'the rock on which the church should be built,' who also obtained 'the keys of the kingdom of heaven...'” Tertullian, On the Prescription Against the Heretics, 22 (c. A.D. 200).
“And Peter, on whom the Church of Christ is built, against which the gates of hell shall not prevail...” Origen, Commentary on John, 5:3 (A.D. 232).
“'...thou art Peter and upon this rock I will build my Church' ... It is on him that he builds the Church, and to him that he entrusts the sheep to feed. And although he assigns a like power to all the apostles, yet he founded a single Chair, thus establishing by his own authority the source and hallmark of the (Church's) oneness...If a man does not fast to this oneness of Peter, does he still imagine that he still holds the faith. If he deserts the Chair of Peter upon whom the Church was built, has he still confidence that he is in the Church?” Cyprian, De Unitate Ecclesiae (Primacy text), 4 (A.D. 251).
“And Peter,on whom the Church of Christ is built, 'against which the gates of hell shall not prevail'” Eusebius, Ecclesiastical History, 6:25 (A.D. 325).
“...the chief of the disciples...the Lord accepted him, set him up as the foundation, called him the rock and structure of the church.” Aphraates, De Paenitentibus Homily 7:15 (A.D. 337).
“[F]or the good of unity blessed Peter, for whom it would have been enough if after his denial he had obtained pardon only, deserved to be placed before all the apostles, and alone received the keys of the kingdom of heaven, to be communicated to the rest.” Optatus of Milevis, De Schismate Donatistorum, 7:3(A.D. 370).
“As I follow no leader save Christ, so I communicate with none but your blessedness, that is with the chair of Peter. For this, I know, is the rock on which the church is built!” Jerome, To Pope Damasus, Epistle 15 (A.D. 375).
“But you say, the Church was rounded upon Peter: although elsewhere the same is attributed to all the Apostles, and they all receive the keys of the kingdom of heaven, and the strength of the Church depends upon them all alike, yet one (Peter) among the twelve is chosen so that when a head has been appointed, there may be no occasion for schism.” Jerome, Against Jovinianus, 1 (A.D. 393).
“Peter, that head of the Apostles, the first in the Church, the friend of Christ, who received revelation not from man but from the Father...this Peter, and when I say Peter, I mean that unbroken Rock, the unshaken foundation, the great Apostle, the first of the disciples, the first called, the first to obey.” John Chrysostom, De Eleemosyna, 3:4 (ante A.D. 407).
Too bad no one taught these early Christians Greek…
Many more quotes can be found at Scripture Catholic.