These are the verses that the Catholic denomination builds their doctrine on: Matthew 16:17-18, "And Jesus answered him. “You are blessed, Simon son of Jonah, because flesh and blood did not reveal this to you, but my Father in heaven! And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overpower it."
Now if you go a few verses after, it says this: Matthew 16:22-23, "So Peter took him aside and began to rebuke him: “God forbid, Lord! This must not happen to you!” But he turned and said to Peter, “Get behind me, Satan! You are a stumbling block to me, because you are not setting your mind on God’s interests, but on man’s.”
Notice that Jesus calls him a) "Satan" and b) a stumbling block. (The latter is obviously in contrast to "on this rock...")
The Catholic church has a clear need to elevate Peter to a position that is not substantiated by Scripture. (The same way that other unScriptural doctrines are promoted.) He was a flawed, vacillating man as is clearly shown in the Bible.
Read Galatians if you want to see Peter for who he actually was, not how you want him to be.