The Roman Catholic Church has infallibly defined the interpretation of Matthew 16.
The Council of Ephesus, 431 AD
No one doubts, in fact, it is obvious to all ages that the holy and most Blessed Peter, head and Prince of the Apostles, the pillar of faith, and the foundation of the Catholic Church, received the keys of the kingdom from our Lord Jesus Christ, the savior and redeemer of the human race.
First Vatican Council, 1870, The First Dogmatic Constitution of the Church of Christ, Chapter 2
Therefore if anyone says that the blessed Apostle Peter was not constituted by Christ the Lord as the Prince of all the Apostles and the visible head of the whole Church militant, or that he received immediately and directly from Jesus Christ our Lord only a primacy of honor and not a true and proper primacy of jurisdiction: anathema sit.
Christ continues with the conferral of the “keys” which appears to be a clear statement of a position of leadership authority.
Mt 16:19-20
I will give you (singular) 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.
This biblical commission echoes one other conferral of keys in the Bible. Eliakim receives the keys of the royal palace.
Is 22:22
I will place the key of the House of David on his shoulder; when he opens, no one shall shut, when he shuts, no one shall open.
Apart from this passage, there is no background in biblical language for binding and loosening. In Rabbinical Judaism, the words signify rabbinical decisions; to bind is to give a decision that imposes an obligation, and to loose is to give a decision that removes an obligation.
Mt 18:15-18
“If your brother sins (against you), go and tell him his fault between you and him alone. If he listens to you, you have won over your brother. If he does not listen, take one or two others along with you, so that ‘every fact may be established on the testimony of two or three witnesses.’ If he refuses to listen to them, tell the church (ekklesia). If he refuses to listen even to the church, then treat him as you would a Gentile or a tax collector. Amen, I say to you, whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”
In Matthew 18:18, the Apostles share in the power to bind and loose that was given to Peter in 16:19; what was given to Peter alone is now shared by the whole Church in the person of the Apostles.
If Peter held a position of primacy, the other Apostles would have to know that and would have reflected that role thrust on Peter by Christ in their relationships to him. In other words, does the Bible reveal a primary place or role for Peter consciously acknowledged by the New Testament writers? Yes, the biblical portrait of Peter presented earlier in this chapter attests to the preeminent role of Peter among the writers of the New Testament.
Among the Apostolic Fathers, the same recognition can be shown.
Tertullian (Rome, 160 - 220 AD), On Monogamy, Chapter 8
Peter alone do I find … to have been married. Monogamist I am led to presume him by consideration of the church, which, built upon him, was destined to appoint every grade of her Order from monogamists.
Clement (Alexandria, 150 - 215 AD), Who Is the Rich Man That Shall Be Saved?, Chapter 21
Therefore, on hearing those words, the blessed Peter, the chosen, the pre-eminent, the first of the disciples, for whom alone and Himself the Savior paid tribute, quickly seized and comprehended the saying. And what does he say? “Lo, we have left all and followed Thee.”
Cyprian (Carthage, 200 - 258 AD), On the Unity of the Catholic Church, Chapter 4
Upon him (Peter), being one, He (Christ) built His Church and although after His resurrection He bestows equal power upon all the Apostles, and says: “As the Father has sent me, I also send you. Receive the Holy Spirit: if you forgive the sins of anyone, they will be forgiven him; if you retain the sins of anyone, they will be retained” (Jn 20:21), that He might display unity, He established by His authority the origin of the same unity as beginning from one.
Cyril (Jerusalem, 315 - 387 AD), Catecheses, No. 2:19
Peter, the chiefest and foremost of the Apostles, denied the Lord thrice before a little maid: but he repented himself, and wept bitterly.
Augustine (Numidia, now Algeria, 354 - 430 AD), Letters, No 53
For, if the order of succession of Bishops is to be considered, how much more surely, truly and safely do we number them from Peter, to whom, as representing the whole Church, the Lord said: “Upon this rock I will build my church and the gates of hell shall not prevail against it” (Mt 16:18). For, to Peter succeeded Linus, to Linus Clement, to Clement Anacletus, to Anacletus Evaristus …