First, the Gospel of Matthew was written in Aramaic, the language Jesus used. In Aramaic, Jesus said in Matthew 16:18
Note that the word for Peter, ke'pha', is the same word for rock. The words are equated: Peter is the rock.'aph 'ena' 'amar-na' lak da'(n)t-(h)uw ke'pha'
and I say - I to thee that-thou-art Kephaswe`'al hade' ke'pha' 'ebneyh le`i(d)tiy
and upon this rock I will build her namely my church
The core of the meaning appears to rest in the two words for a "rock." If Matthew recorded that Christ used the same word both for (1) the proper name of Peter and (2) the foundation on which Christ says he will build the church, then an interpretation follows that the foundation of the church is Peter.
Karl Keating explains further:
Beyond the grammatical evidence, the structure of the narrative does not allow for a downplaying of Peters role in the Church. Look at the way Matthew 16:15-19 is structured. After Peter gives a confession about the identity of Jesus, the Lord does the same in return for Peter. Jesus does not say, "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 an insignificant pebble and on this rock I will build my Church. . . . I will give you the keys of the kingdom of heaven." Jesus is giving Peter a three-fold blessing, including the gift of the keys to the kingdom, not undermining his authority. To say that Jesus is downplaying Peter flies in the face of the context. Jesus is installing Peter as a form of chief steward or prime minister under the King of Kings by giving him the keys to the kingdom. As can be seen in Isaiah 22:22, kings in the Old Testament appointed a chief steward to serve under them in a position of great authority to rule over the inhabitants of the kingdom. Jesus quotes almost verbatum from this passage in Isaiah, and so it is clear what he has in mind. He is raising Peter up as a father figure to the household of faith (Is. 22:21), to lead them and guide the flock (John 21:15-17). This authority of the prime minister under the king was passed on from one man to another down through the ages by the giving of the keys, which were worn on the shoulder as a sign of authority. Likewise, the authority of Peter has been passed down for 2000 years by means of the papacy.
God bless.
Not original, I persume, but a translation from the Greek.