The passage you are asking about occurs only in Matthew 16:17-19. Both Mark and Luke record Peter's recognition of Jesus as the Christ, which immediately precedes this passage; but they do not record Jesus' response: "Blessed are you, Simon Bar-Jona! For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are Peter, and on this rock I will build my church, and the powers of death shall not prevail against it. I give you the keys to the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven."
As you noted in your question, the Roman Catholic Church cites this passage as proof that Peter—who later became the first Bishop of Rome—was appointed by Jesus to head the church, and that that appointment extends to all future Bishops of Rome—whence comes the primacy of the Pope. From their perspective, that makes sense.
From the perspective of the message and ministry of Jesus, it isn't quite so clear-cut. The word translated as 'church' didn't suggest a structured organization, but simply "the people of God." The renaming of Simon carries forward a long biblical tradition that a breakthrough in spiritual awareness be accompanied by a new name (Abram became Abraham, Jacob became Israel, etc.). And, of course, it's also a play on words, since in both Aramaic and Greek the proper name (Kepha and Petra, respectively) is almost identical to the word for "stone." So essentially: "Your new name is Rock, and on this Rock ..."
Metaphysically, I think what has happened here is that Simon has had a dramatic spiritual breakthrough, and Jesus is deeply moved as he witnesses it. In watching the unfolding ministry of Jesus of Nazareth, Peter is suddenly aware that he is seeing, not just a gifted teacher, but the Christ—the Logos, the Word, the creative Power of God—in expression. It is really Jesus' first tangible "victory," since his entire purpose is to make people aware of the Christ expressing as him, so that they can go on to find and express the same Christ energy within themselves. He realizes that Peter's sudden awareness is a divine gift—"flesh and blood has not revealed this to you, but my Father who is in heaven"—that merits the bestowal of a new name.
It is not—again, as I understand the passage—the man Simon Peter who is to be "the rock" on which God's people will be supported. It is our ability to understand and embrace the Christ that will be the solid foundation under our continued spiritual growth. As we recognize and embrace our own Christ nature, and begin to express that nature, then life will work very differently. We will understand that we are "bound" or "free" in our earthly experiences according to the choices we make in consciousness.