The previous postings on this topic have all been great. I just want to add one more perspective, to elaborate a bit more on the second part of of Iacov's question.
Brett was absolutely right in his answer, that Christ's statement had nothing to do with Peter *himself* but with his *confession of faith* in Christ as the Son of God. In the original Greek, when Christ says "You are Petros (Peter) and on this rock...." -- the "this rock" is in the feminine case, which it could not have been if the subject were the masculine noun "Petros" or the person of Peter himself. In fact, the rock referred to was the rock of faith that Peter had confessed.
Another point that the Roman Catholics often use to defend the primacy of the Pope as the successor to Peter, is when Christ asks the Apostle three times if he loves Him, and tells him "feed My sheep." They take that as Christ bestowing a special power or role on Peter. But actually quite the reverse is true. All Christ was doing at that point was *restoring* Peter to the apostleship, after Peter had denied Christ thrice. It is interesting to note that, immediately after the Resurrection, when the angels speak to the myrrh-bearing women, they tell them to go and announce the good news of Christ's resurrection to "the Apostles and Peter". At that point, Peter was still sort of "outside" of the group of the Apostles because of his triple denial of Christ. Hence, Christ's question to and reinstatement of Peter to his former position among the Apostles -- not to bestow on him any kind of primacy, but simply to bring him back into the fold.
I hope this helps!
In Christ,
Damian