Protestants typically interpret this verse to be referring to Christ, the Chief Cornerstone, when he speaks of the rock upon which the church will be built.
One reason for this is that Peter as a proper name for Simon is masculine in form -- petros (Strong's G4074). When Jesus says "on this rock", the word for "rock" is feminine -- petra (Strong's G4073).
The distinction according to Strong's is as follows:
G4073 — Strong
πέτρα - petra - pet'-ra
Feminine of the same as G4074; a (mass of) rock (literally or
figuratively): - rock.
G4074 — Strong
Πέτρος - Petros - pet'-ros
Apparently a primary word; a (piece of) rock (larger than G3037); as a
name, Petrus, an apostle: - Peter, rock. Compare G2786.
So, there does seem to be a difference used in the words for rock:
You are **Peter**, **a piece of rock**, and upon this **mass of rock** I will build my church.
Peter's Own Reference to Christ as the Rock
In 1 Peter 2, Peter himself refers to Christ as the Rock:
And coming to Him as to a living stone which has been rejected by men, but is choice and precious in the sight of God, 5 you also, as living stones, are being built up as a spiritual house for a holy priesthood, to offer up spiritual sacrifices acceptable to God through Jesus Christ. 6 For this is contained in Scripture:
“Behold, I lay in Zion a choice stone, a precious corner stone,
And he who believes in Him will not be disappointed.” [Isaiah 28:16]
7 This precious value, then, is for you who believe; but for those who disbelieve,
“The stone which the builders rejected, This became the very corner stone,” [Psalm 118:22]
“A stone of stumbling and a rock of offense”; [Isaiah 8:14]
for they stumble because they are disobedient to the word, and to this doom they were also appointed. 1 Peter 2:4-8 NASB
So, here Peter refers to Isaiah and the Psalms that point to a particular Rock or Stone, which is Jesus--not Peter. It is Jesus who is the Chief Cornerstone for the foundation of the church. It is Jesus on whom we must believe. It was Jesus who was rejected. And it was Jesus who was a stone of stumbling.
Finally, in Nebuchadnezzar's dream recorded in Daniel chapter 2, there is a "a stone was cut out without hands" which struck the statue, which represented the four world empires and then itself became a kingdom that filled the earth.
You continued looking until a stone was cut out without hands, and it struck the statue on its feet of iron and clay and crushed them. 35 Then the iron, the clay, the bronze, the silver and the gold were crushed all at the same time and became like chaff from the summer threshing floors; and the wind carried them away so that not a trace of them was found. But the stone that struck the statue became a great mountain and filled the whole earth... Daniel 2:34-35 NASB
In the days of those kings the God of heaven will set up a kingdom which will never be destroyed, and that kingdom will not be left for another people; it will crush and put an end to all these kingdoms, but it will itself endure forever. 45 Inasmuch as you saw that a stone was cut out of the mountain without hands and that it crushed the iron, the bronze, the clay, the silver and the gold, the great God has made known to the king what will take place in the future; so the dream is true and its interpretation is trustworthy.” Daniel 2:44-45 NASB
So, in Nebuchadnezzar's dream, Jesus is the Stone/Rock.
The Rock of Moses
Moses also struck a rock on two different occasions to provide water for the people in the desert. This rock is symbolic of Jesus as well. Moses was commanded to strike the rock the first time, but to merely speak to it the second time. This symbolized Christ being struck once for all, but never again. (Exodus 17 and Numbers 20)
Many other references
The Old Testament has quite a few other references to the Rock, which appears to be a title for the Messiah. Here are just a few:
The God of Israel said, The Rock of Israel spoke to me, ‘He who rules over men righteously, Who rules in the fear of God, 2 Samuel 23:3 NASB
The Lord is my rock and my fortress and my deliverer, My God, my rock, in whom I take refuge; My shield and the horn of my salvation, my stronghold. Psalm 18:2 NASB
He only is my rock and my salvation, My stronghold; I shall not be greatly shaken. Psalm 62:2 NASB
- Peter is distinguished as a separate kind of rock than Christ in Matthew 16.
- Peter himself refers to Christ as the Rock/Chief Cornerstone in fulfillment of writing in Isaiah and the Psalms.
- Daniel refers to a rock that is fulfilled by Jesus.
- The rock that Moses struck symbolizes Jesus.
- Many other places in the Old Testament refer to God as the Rock.
For all these reasons, the Matthew 16 passage is understood, to exalt Jesus, then, as the Rock/Chief Cornerstone, rather than Peter as the head of the church (pope).