Matthew 16:18 And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.
I want to first dispel a misunderstanding on which the Roman Catholic Church was founded. Roman Catholics are taught that the church was founded on the rock which was Peter. This is incorrect. The word Peter is a transliteration of the Greek petros, "stone". The word "rock" is a translation of the Greek petra, which means great rock, quarry. Clearly what Jesus was saying to Simon when He named him Peter was that Simon had now become a stone in the edifice that He was going to build. What then was the Rock on He would build it?
The Bible says "and the Rock was Christ" (1 Corinthians 10:4). We can imagine that Jesus pointed to Himself when He said "upon this Rock" I will build My church." Such a gesture would not find its way into the translation of His words.
We can also say that the Rock was the confession that Simon had just made in verse 16. Let's read that verse in context:
Matthew 16:15 He saith unto them, But whom say ye that I am?
16 And Simon Peter answered and said, Thou art the Christ, the Son of the living God.
17 And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven.
18 And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.
Flesh and blood had not revealed what? The revelation that the Father had just made to Simon Barjona (son of Jonah), and which he had just confessed.
In context, "upon this rock" Jesus woud build His church.
On the other hand, Peter, always humble, never exalting himself above other members of the body, called all believers "living stones" (1 Peter 2:4,5), using the word "petroi" (petroi is the plural of petros, and thus means "stones").
Now to the main point.
What Jesus was saying that He was going to build upon Peter's confession (of Jesus being the Son of God) a church (Greek, ekklesia), which would be indestructible, which all the forces of hell could not prevail against.
Does this refer to the church as understood in common usage?
It refers to the ekklesia, the invisible church, known only to Himself, as the first-born, the church whose members' names are written, not in any earthly church register, but in the book that is in heaven (Luke 10:20), even the Lamb's Book of Life (Revelation 21:27).