In the creed we profess “One Holy catholic and Apostolic” Church. There's a couple of things mean by the "Apostolic Church" and the idea of Apostolic Succession. The first is that there is an unbroken line from our current bishops back to the original apostles. As Peter, the first Bishop of the Church, consecrated successors, and so on, that chain reaches through the ages until today. When a bishop lays hands on you at your confirmation, you are receiving that laying on of hands passed down over the ages back to those original twelve. The second aspect is that there is an assumption of the faith, teaching, and discipline of the apostles being passed on by Peter and his successors.

For another perspective of the Apostolic Church, St. Irenaeus (a peer of Polycarp, who himself was a student of John) writes:

“the greatest and most ancient church…which was established at Rome by the most renowned Apostles Peter and Paul. This tradition the church has from the Apostles, and this faith has been proclaimed to all men, and has come down to our own day through the succession of bishops. For this church has a position of leadership and authority; and therefore every church, that is, the faithful everywhere, must needs agree with the church at Rome, for in her the apostolic tradition has ever been preserved by the faithful from all parts of the world.