Some areas of agreement between Protestants and Catholics include the fact that Jesus is giving Simon a previously unheard of name, Peter. The English word "Peter" comes from the Greek word petra, which means "rock." Other than the three men and one squirrel pictured on the left, no one goes by a name even remotely similar anymore in our day either. Jesus may as well have named Simon "Ball Point Pen," for Protestants and Catholics will agree that this name bears about as close a resemblance to Simon's actual character as "Rock" did.
Now to our disagreements.
The Protestant will point out that two different words for "rock" are used in Matt. 16:18, which calls into question whether Peter himself is the rock on which the church would be built: "You are Petros, and on this petra I will build my church." Furthermore, Peter goes on in his first epistle to speak of all believers as "living stones" being built up into a spiritual house, the church. This being the case, many Protestants have argued that though Simon is renamed Petros ("little stone"), it is on the petra ("large rock") of his confession of Christ that the church would be built.
It seems to follow from this interpretation that, since the foundation of the church is a common confession of Christ shared by all faithful Christians, the church that is built upon this confession is primarily invisible (though it is manifested in visible churches comprised of professing believers).
The Catholic would respond to the Protestant by saying that though Matthew used two different Greek words for "rock" in v. 18, Jesus didn't speak Greek, he spoke Aramaic. This being the case, what he actually said would have sounded like this: "You are Kepha, and on this kepha I will build my church." Matthew, who wrote his gospel in Greek 30 or 40 years later, needed to employ two different words because Greek, unlike Aramaic, assigns gender to nouns. "Rock" (petra) is feminine, but Simon is a man, hence the change to Petros (it would be like a new father wanting to honor his own dad -- Joseph -- by naming his child after him, but then finding out his newborn is a girl and therefore naming her Josephine). The Catholic would argue, therefore, that Jesus' point would have crystal clear to those who heard him: Jesus would build his church on Peter.
As with the Protestant interpretation of this text, the Catholic understanding has direct ramifications for ecclesiology. If the church is built on Peter himself, it would seem to follow that the church is a visible entity just like Peter was. Assuming for the time being that this arrangement would outlast Peter's earthly life, it would follow that the church that Jesus promised to build would have to be connected to Peter in some way.
OK then, let's hash it out....