And Jesus answering said to him: "And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it. And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven." (Matt. 16:18-19)
PETER IS THE VISIBLE HEAD OF THE CHURCH AND THE VICAR OF CHRIST UPON EARTH: Our Lord built His Church upon the rock of Peter (on the rock which is Peter), giving to it an invincible strength and stability. Thus Peter, according to the will of the Divine Architect, is the immovable foundation of the Church. On him the visible Church, and all its pillars and stones (or to speak without figure, all its members) must, mediately or immediately, rest, and by this support be kept together. Whatever does not rest on this foundation, does not belong to the Church of Christ. We must, therefore, accept the fact that Peter occupies quite a unique position in the Church; that he is its supreme head, and that his office is to keep all other members in the unity of the faith, and that he is, in fact, the supreme authority in the Church. This is to be understood by the power of the keys, which was given to Peter. The giving up to another of the keys of a house is understood by all to be the token of the surrender to that person of supreme authority over the house and its inhabitants. So, when our Lord Jesus gave to St. Peter the power of the keys, He gave him supreme authority over His Church, that is, the authority to teach, to judge, and to make laws. The promise which our Lord on this occasion made to Peter, He fulfilled after His resurrection, when He have him the office of Chief Pastor. By this supreme authority Peter became the visible representative of our Lord Jesus Christ upon earth, and this Primacy of St. Peter must needs continue so long as the Church lasts; in other words, there must be successors of Peter.
[From 'A Practical Commentary on Holy Scripture' by Bishop Knecht, D.D.]
(1899 Douay-Rheims Bible)