JOEL - Here are the scriptures in question. Jesus said:

13. Whom do men say that I the Son of man am?
14. And they said, Some say that thou art John the Baptist: some, Elias; and others, Jeremias, or one of the prophets.
15. He saith unto them, But whom say ye that I am?
16. And Simon Peter answered and said, Thou art the Christ, the Son of the living God.
17. And Jesus answered and said unto him, Blessed art thou, Simon Bar-jona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven.
18. And I say also unto thee, That thou art Peter; and upon this rock I will build my church; and the gates of hell shall not prevail against it.
19. And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.(Matthew 16:13-19)

Luke tells us that the first apostle the Savior chose was Simon (Luke 6: 13-14), the son of Jona. Simon, son of Jona, was the name by which he was best known among his associates, but Christ called him Cephas (John 1: 42, Luke 6:14), which in the Hebrew means rock or stone, and in the Greek the meaning of stone or rock is Petras, or Peter.
In verse 18 above, Jesus again identifies Simon Bajona as Peter. The Catholics conclude that, because Jesus gave him the new name of Peter, the "rock" He is talking about upon which His Church is built, must be Peter and therfore claim him as the first leader or Pope of the Christian Church.
Peter no doubt took a leadership position in the Church in the beginning, however, it is contrary to reason to think that the Lord would establish his Church upon any one man, no matter how faithful and wonderful he might be. It is the Church of Jesus Christ, not the church of Peter. Jesus has also been identified as a rock in the scriptures(1 Cor. 10: 4, Eph 2: 20).

The scriptures tell us that the early Church was not built on one Apostle it was built on a foundation of all the Apostles who lead the Church through revelation from Jesus. Speaking to the saints in Ephesus Paul said:
"Now therefore ye are no more strangers and foreigners, but fellowcitizens with the saints, and of the household of God; And are built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone;" (Eph 2: 19-20)

Here's another way to look at those scriptures in Matthew. In verses 16 and 17 Jesus talks about how it was through revelation that Peter could know that Jesus is the Christ. Then in the next verse He says upon this rock He will build his church. It is quite obvious that the rock He was speaking of is the revealtion from God, that he spoke of in the previous verses. And this is exactly what was used through the prophet Joseph Smith and others to build and maintain The Church of Jesus Christ of Latter-day Saints, just as Jesus said He would do. And we have also obtained those "keys"(verse 19) He talked about, when the priesthood of God was restored through the prophet Joseph Smith, who received it directly from Peter, James, and John.(JSH 1:72)

The Prophet Joseph Smith explained:
"John was a priest after the order of Aaron, and he held keys of the priesthood, and came forth preaching repentance and baptism, for the remission of sins, but at the same time cries out, "There cometh one mightier than I after me, the latchet of whose shoes I am not worthy to stoop down and unloose," and Christ came according to the words of John, and he was greater than John, because he held the keys of the Melchizedek Priesthood and kingdom of God, and had before revealed the priesthood to Moses; yet Christ was baptized by John to fulfill all righteousness; and Jesus in his teaching says: "Upon this rock I will build my Church, and the gates of hell shall not prevail against it." What rock? Revelation.(Teachings of the Prophet Joseph Smith, pp. 273-74. )