The signs of authority that Jesus gave to St. Peter were: Jesus promised Peter a special position in the Church (in the "Rock of My Church" dialogue in Matthew 16:18).
Jesus tells him he will give him the keys of the kingdom of heaven. This means that he is giving him authority.
In the Last Supper, Jesus washes St. Peter's feet first. He carries a long dialogue with Peter. This is a gesture of respect.
The role of St. Peter in the Catholic Church is he is counted as the first pope of Rome and by Eastern Christian tradition he is the first Patriarch.
