The concept of an Office is essential in understanding the role of Peter and his successors in the Petrine office. If we look at the story of Christ, Peter, and the Keys in Mt. 16, the symbol handed on by Christ is that of the Keys to the Kingdom. This hearkens back to Eliakim (in Isaiah), who was the chief officer in the Davidic Kingdom, and who carried actual keys. This office was passed on to another upon his death.
In the new Davidic Kingdom, Christ chose one Apostle to serve as the chief officer, the new keybearer... that keybearer was Simon, whom Christ gave the name of Peter. Upon his death, a successor was chosen, and held the same authority among the remaining Apostles and their successors that Peter held among the Twelve. That authority is one where whatever he binds on earth is bound in heaven, and what he loosens on earth is loosened in heaven.