'On this rock'
'On this rock I will build my church' — Peter, Peter's confession, or Christ; the keys and binding as papal, synodal or the whole church's.
Matthew 16:13-19
13Now when Jesus came into the parts of Caesarea Philippi, he asked his disciples, saying, “Who do men say that I, the Son of Man, am?”
14They said, “Some say John the Baptizer, some, Elijah, and others, Jeremiah, or one of the prophets.”
15He said to them, “But who do you say that I am?”
16Simon Peter answered, “You are the Christ, the Son of the living God.”
17Jesus answered him, “Blessed are you, Simon Bar Jonah, for flesh and blood has not revealed this to you, but my Father who is in heaven.
18I also tell you that you are Peter, and on this rock I will build my assembly, and the gates of Hades will not prevail against it.
19I will give to you the keys of the Kingdom of Heaven, and whatever you bind on earth will have been bound in heaven; and whatever you release on earth will have been released in heaven.”
What each tradition says
Rabbinic / Midrashic
Kaufmann Kohler reads Matthew 16:19 and 18:18 as Jesus using a familiar formula when appointing his disciples as his successors. By these words, he says, Jesus virtually invested them with the same authority the scribes and Pharisees had to bind heavy burdens and to loose them.
Kaufmann Kohler (1843-1926), The Jewish Encyclopedia (New York: Funk & Wagnalls, 1901-1906), s.v. 'Binding and Loosing' — Section 'In the New Testament' (on Matt. xvi. 19, xviii. 18)
Patristic
Chrysostom says that “on this rock” means on the faith of his confession. Christ thereby signifies that many were now on the point of believing, raises his spirit, and makes him a shepherd.
John Chrysostom (c. 349-407), Homilies on the Gospel of St. Matthew, in Nicene and Post-Nicene Fathers, ser. 1, vol. 10 (ed. Philip Schaff; New York: Christian Literature Publishing Co., 1886) — Homily LII, section 3 (on Matthew 16:18)
Patristic
Cyprian says Peter was first chosen by the Lord and was the one upon whom He built His Church. Yet when Paul later disputed with him about circumcision, Peter did not insolently claim anything to himself or arrogantly assume anything.
Cyprian of Carthage (c. 200-258), Epistle LXX, To Quintus, Concerning the Baptism of Heretics, in Ante-Nicene Fathers, vol. 5 (eds. Alexander Roberts and James Donaldson; Buffalo: Christian Literature Publishing Co., 1886) — Epistle LXX, section 3
Catholic
George Joyce argues that in the original Aramaic the word for Peter and the word for rock are one and the same. Therefore, attempts to explain "rock" as referring not to Peter himself but to something else are misinterpretations. It is Peter who is the rock of the Church.
George Joyce, The Catholic Encyclopedia, vol. 12, s.v. 'The Pope' (New York: Robert Appleton Company, 1911) — Section 'Primacy of Peter' / the exegesis of Matthew 16:18-19
Orthodox
No sourced position found. Checked: John of Damascus, An Exact Exposition of the Orthodox Faith (NPNF ser. 2, vol. 9) — 'Thou art Peter' occurs nowhere in the volume; The Acts and Decrees of the Synod of Jerusalem (1672, trans. J. N. W. B. Robertson, 1899, OCR of the archive.org scan) — the only 'rock' is a bookseller's name; The Longer Catechism of Philaret (in The Doctrine of the Russian Church, trans. R. W. Blackmore, 1845) — the word 'rock' does not occur; the 1894 Moscow Catechism of the Orthodox Eastern Church — no treatment of the passage; no public-domain English text of the 1848 Encyclical of the Eastern Patriarchs (the standard Eastern answer on the primacy) was found on archive.org.
Reformed / Protestant
Calvin says “this rock” shows the name Peter applies both to Simon individually and to other believers: they are founded on Christ’s faith and joined by holy consent into a spiritual building where God dwells. Christ announced this as the whole Church’s common foundation and intended to associate all the godly who would ever exist with Peter.
John Calvin (1509-1564), Commentary on a Harmony of the Evangelists, Matthew, Mark, and Luke, vol. 2 (trans. William Pringle; Edinburgh: Calvin Translation Society) — Commentary on Matthew 16:13-19, on verse 18
Evangelical
Barnes states it is not said here or anywhere else in the Bible that Peter should have infallible successors as Christ's vicegerents and head of the church. The meaning, he says, is that Christ would make Peter the honored instrument of first making known his gospel to Jews and Gentiles, and a firm and distinguished preacher in building his church.
Albert Barnes (1798-1870), Notes on the New Testament, Explanatory and Practical (enlarged ed. by Robert Frew; CCEL carries the Baker reprint) — Note on Matthew 16:18