The Roman Catholic interpretation of these verses is that Jesus was saying, "Peter, your name means a stone (John 1:42) and upon you I will build my church." They allege that Peter was the first pope at Rome and that his authority is passed on to the popes who follow by succession. They teach that tradition and the decrees of the pope are of equal weight with the Bible and that whatever is bound or loosed by "the Church" will dictate that which is bound or loosed in heaven.
The typical Protestant interpretation is that Jesus was saying, "Peter, the truth that you have spoken is the rock upon which I will build my church." Thus, it is allowed that all who believe and confess that Jesus is "the Christ, the Son of the living God" are in a universal invisible church.
The Baptist interpretation is that Jesus was declaring that He, Himself, was the rock upon which He would build His church. There are many verses throughout Exodus, Numbers, Deuteronomy, 1st and 2nd Samuel, the Psalms, and Isaiah that refer to Christ and to God as "the Rock" and as "my rock." In Exodus 17 and Numbers 20, the rock in the wilderness is seen as a type of Christ. That rock in the wilderness is referred to in 1 Corinthians 10:4 which says that "that Rock was Christ." The prophecy of Isaiah 8:13-15 and 28:16 refer to Christ as "a stone of stumbling," a "rock of offence," a "precious corner stone," and "a sure foundation," as is verified by Romans 9:33. Matthew 21:42 says:
Psalm 118:22-23 says:
1 Peter 2:5-8 describes a church as "lively stones" that "are built up a spiritual house," with Jesus being the "chief corner stone." Ephesians 2:20-22 speaks of a church as being "built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone":
It is essential that "the house of God, which is the church of the living God, the pillar and ground of the truth," (1 Timothy 3:15) be built upon a firm foundation. Jesus said, ". . . upon this rock I will build my church." That Rock was Christ.