Introduction

Infallibility is often raised by critics to attack the Church because it is easy to distort for purposes of ridicule. That the Church is infallible in Her definitions on faith and morals has always been held as true. To be infallible is to be protected from error. It does not allow the Church to create new truths. All that has ever been infallibly declared has previously been divinely revealed. Infallibility is not impeccability, nor inspiration. This gift never depends on the merits of the individual or even the fallible arguments supporting a teaching. Only the teaching itself is protected by Christ’s promise. The organs of the Church capable of such a decree include the Pope teaching to the whole church ex cathedra, or by decree of an ecumenical council. If it’s not on faith and morals, or not binding on the whole Church, it’s not protected.

Catechism
"The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when, as supreme pastor and teacher of all the faithful - who confirms his brethren in the faith he proclaims by a definitive act a doctrine pertaining to faith or morals. . . . The infallibility promised to the Church is also present in the body of bishops when, together with Peter's successor, they exercise the supreme Magisterium," above all in an Ecumenical Council. When the Church through its supreme Magisterium proposes a doctrine "for belief as being divinely revealed, “and as the teaching of Christ, the definitions "must be adhered to with the obedience of faith." This infallibility extends as far as the deposit of divine Revelation itself.
- Catechism of the Catholic Church, 2nd ed., (1997), n.891
Scripture

Isa. 35:8 - In the Old Testament, the Church is described as a Holy Way, free from error.

Acts. 9:2; Acts. 22:4; Acts. 24:14,22 - In the New Testament, the Church is the "Way" from Isaiah where fools will not err therein.

Mt. 16:17 – God protects Peter from error by revealing the truth in a doctrinal question of faith.

Mt. 16:18 - If the gates of hell shall not prevail, then the church cannot lose the faith. It cannot teach error. It is infallible.

Mt. 16:19 – The power to bind and loose on earth must be infallible for the sake of Divine Justice in heaven.

Mt. 18:17-19 - Jesus exhorts people to seek the Church as a final authority on the truth.

Lk. 22:32 - Jesus prays for Peter alone to be protected from Satan (an opposer of Truth). This implies infallibility.

Jn. 16:13; Mt. 28:20 – Matthew and John say God will guide the Church into all truth.

1 Tim. 3:15 - Paul says the Church is the pillar and foundation of the truth.

1 Pet. - 2 Pet. – Peter pens two infallible encyclicals in the canon of scripture.

Tradition & Fathers
"For where the Church is, there is the Spirit of God; and where the Spirit of God is, there is the Church, and every kind of grace; but the Spirit is truth."
- St. Irenaeus of Lyons, Against Heresies (A.D. 130-202)
“Suppose now that all [the Churches] have erred; that the apostle was mistaken in giving his testimony; that the Holy Ghost had no such respect to any one (church) as to lead it into truth, although sent with this view by Christ, and for this asked of the Father that He might be the teacher of truth; suppose, also, that He, the Steward of God, the Vicar of Christ, neglected His office, permitting the churches for a time to understand differently, (and) to believe differently, what He Himself was preaching by the apostles—is it likely that so many churches, and they so great, should have gone astray into one and the same faith?”
- Tertullian of Carthage, Prescription against Heretics (A.D. ~200)
"The spouse of Christ cannot be adulterous; she is uncorrupted and pure.”
- St. Cyprian of Carthage, De Unitate Ecclesiae (A.D. 251)
"But the word of the Lord which came through the ecumenical Synod at Nicaea, abides forever."