That's where -- unlike in the same story told in Mark and Luke -- Jesus says, "You are Peter, and on this rock I will build my church." One Catholic reading is that Jesus meant to break away from Judaism, start a new religion and name Peter the first pope.
Many other Catholics and Protestants, including me, conclude that Jesus, always a Jew, was much more interested in announcing the in-breaking reign of God (which is what Jesus meant by the Gospel) and was not focused on creating a hierarchical and institutionalized structure. Indeed, it's hard to find a biblical scholar these days who would agree that Jesus' purpose was to start a new religion.
Still, I've never quite known how to respond to insistent Catholics who haul out Matthew 16:18 and assume that's a full answer that confirms Petrine primacy and, by extension, the assertion that anyone outside the Catholic church is apostate.
I've just received considerable help with this matter from a Capuchin Franciscan, Michael H. Crosby, in his book Repair My House.
Crosby takes a careful look at Matthew 16, comparing it to the Mark 8 and Luke 9 versions of the same story, versions that omit Jesus saying anything to Peter about being the rock (a play on his name) on which he will build a church.
To use the Matthew 16:18 passage as the be-all of Catholic reliance on Scripture and tradition, he writes, is to lean on "a selective and fundamentalist interpretation of this one text." Doing that, he says, "represents both intellectual dishonesty and scriptural errancy."
A more balanced and satisfying way of understanding the role of the church and its leadership, Crosby writes, is to balance Matthew 16 with Matthew 18. In the latter chapter, we find a broader notion of authority among the followers of Jesus, disciples who eventually would separate from Judaism and become Christianity.
In Matthew 18, Jesus gives authority and responsibility not to a single person but to the whole church (ekklesia, in Greek, which means literally the community that has been called out). Verse 17 says that a member of the ekklesia is to report to the ekklesia if someone who has sinned against him or her will not repent.
Thus, Crosby notes, Jesus assigns to the local church in 18:18 the power to bind and loose that is given to Peter in 16:19, and "both texts must be considered as equal in their power to bind and loose." The difference is that Peter gets the "keys to the kingdom" in 16 while the community in 18 "receives the promise of Christ's abiding presence in their binding and loosing in a way that is not given to Peter."