Jesus told Peter, "upon this rock I will build my church." The Catholics believe the rock was Peter; however, in context, the rock of Israel was Christ. Jesus had also queried Peter as to his own identity, and Peter had answered under the inspiration of the Holy Spirit, "Thou art the Messiah, the Son of the living God." To which the Lord replied, "Flesh and blood hath not revealed this unto thee, but my Father which is in Heaven." It was then that the Lord said, "upon this rock" he would build his church, "and the gates of Hell shall not prevail against it."
The "it" here is clearly "the church." The sure knowledge Peter received was revelation from God. It's entirely likely that the church also is based upon the "rock" of revelation -- or the spiritual knowledge that comes, not through the knowledge of man but by the witness of God. Jesus additionally said, "and the gates of Hell shall not prevail against it." This is frequently cited by the Catholics and others to show that the church would "prevail" through the ages and would never cease. But John talks about the church being led into the wilderness for 1260 days (years), and the scripture says it's not the church that's the subject, but the "gates of hell" that would not prevail against the church. In other words, the church is assaulting the gates of Hades, or death. These gates would resist, but they would not prevail against the church (see Matthew 16:18). The very next verse speaks of the keys of authority Peter would need.
"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." These keys ensure the proper administration of the church so that its ordinances would apply on Heaven and on Earth.
So as I see it, if Jesus has a church, it will be like his ancient church. It will have his name, be guided by him and have his authority. It also will be built like the ancient church, with apostles, prophets, bishops, elders, priests, deacons and other offices. And it will be guided by revelation and have an open canon of scripture, just as the ancient church.