“He who hears you hears me, and he who rejects you rejects me, and he who rejects me rejects him who sent me.”
Short answer
Christ sent the apostles to teach all nations and promised to be with them to the close of the age. The Church teaches that their successors, the bishops in communion with the Pope, have the task of interpreting the word of God with authority. This teaching office is not above the word of God but serves it, and it accepts no new public revelation. No individual bishop is infallible. The bishops teach infallibly only when, united with one another and with the successor of Peter, they agree that a point of faith or morals is to be held definitively.
Drafted from primary sources. Every quotation is checked against the page it links to. Papal infallibility has its own timeline.
Where this teaching appears on the Offshoots timelines: August 1520, Scripture judged by each believer, not by the pope; 18 April 1521, Luther at the Diet of Worms; 1571, The Thirty-nine Articles.
16 quotations in date order
“If he refuses to listen to them, tell it to the church; and if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector. Truly, I say to you, whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”
“All authority in heaven and on earth has been given to me. Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you; and lo, I am with you always, to the close of the age.”
“The apostles and the elders were gathered together to consider this matter. … For it has seemed good to the Holy Spirit and to us to lay upon you no greater burden than these necessary things”
“…the household of God, which is the church of the living God, the pillar and bulwark of the truth.”
“For we ought to receive every one whom the Master of the house sends to be over His household, [Matthew 24:45] as we would do Him that sent him. It is manifest, therefore, that we should look upon the bishop even as we would upon the Lord Himself.”
“Wherefore it is incumbent to obey the presbyters who are in the Church — those who, as I have shown, possess the succession from the apostles; those who, together with the succession of the episcopate, have received the certain gift of truth, according to the good pleasure of the Father.”
“For my part, I should not believe the gospel except as moved by the authority of the Catholic Church.”
“But here some one perhaps will ask, Since the canon of Scripture is complete, and sufficient of itself for everything, and more than sufficient, what need is there to join with it the authority of the Church's interpretation? For this reason — because, owing to the depth of Holy Scripture, all do not accept it in one and the same sense, but one understands its words in one way, another in another; so that it seems to be capable of as many interpretations as there are interpreters.”
“Why does he say Though we? Why not rather though I? He means, though Peter, though Andrew, though John, in a word, though the whole company of apostles, preach unto you other than we have preached unto you, let him be accursed. Tremendous severity! He spares neither himself nor his fellow apostles, so he may preserve unaltered the faith which was at first delivered.”
“The universal Church cannot err, since she is governed by the Holy Ghost, Who is the Spirit of truth: for such was Our Lord's promise to His disciples [John 16:13]: "When He, the Spirit of truth, is come, He will teach you all truth."”
“…no one, relying on his own skill, shall,–in matters of faith, and of morals pertaining to the edification of Christian doctrine, –wresting the sacred Scripture to his own senses, presume to interpret the said sacred Scripture contrary to that sense which holy mother Church,–whose it is to judge of the true sense and interpretation of the holy Scriptures,–hath held and doth hold”
“The entire body of the faithful, anointed as they are by the Holy One, cannot err in matters of belief. They manifest this special property by means of the whole peoples' supernatural discernment in matters of faith when "from the Bishops down to the last of the lay faithful" … they show universal agreement in matters of faith and morals. That discernment in matters of faith is aroused and sustained by the Spirit of truth. It is exercised under the guidance of the sacred teaching authority, in faithful and respectful obedience to which the people of God accepts that which is not just the word of men but truly the word of God.”
“Although the individual bishops do not enjoy the prerogative of infallibility, they nevertheless proclaim Christ's doctrine infallibly whenever, even though dispersed through the world, but still maintaining the bond of communion among themselves and with the successor of Peter, and authentically teaching matters of faith and morals, they are in agreement on one position as definitively to be held. … And this infallibility with which the Divine Redeemer willed His Church to be endowed in defining doctrine of faith and morals, extends as far as the deposit of Revelation extends, which must be religiously guarded and faithfully expounded. … but a new public revelation they do not accept as pertaining to the divine deposit of faith.”
“But the task of authentically interpreting the word of God, whether written or handed on, has been entrusted exclusively to the living teaching office of the Church, whose authority is exercised in the name of Jesus Christ. This teaching office is not above the word of God, but serves it, teaching only what has been handed on, listening to it devoutly, guarding it scrupulously and explaining it faithfully in accord with a divine commission and with the help of the Holy Spirit, it draws from this one deposit of faith everything which it presents for belief as divinely revealed.”
“It is this Magisterium's task to preserve God's people from deviations and defections and to guarantee them the objective possibility of professing the true faith without error. Thus, the pastoral duty of the Magisterium is aimed at seeing to it that the People of God abides in the truth that liberates. To fulfill this service, Christ endowed the Church's shepherds with the charism of infallibility in matters of faith and morals.”