From the first post, the Church teaches solemnly and by her ordinary and universal Magisterium via her Teaching Authority, her Magisterium, that is, the Pope and the Bishops in communion with the Pope.
One of her teachings is that she cannot err when and in what she teaches a doctrine of faith or morals. This is what is meant by the infallibility of the Church.
What is infallibility
In general, infallibility is exemption or immunity from liability to error or failure; in particular in theological usage, as regards the Church, it is the supernatural prerogative by which the Church of Christ is, by a special Divine assistance, preserved from liability to error when and in what she teaches a doctrine of faith or morals.
For what purpose is the charism of infallibility Christ endowed his Church with?
CCC 889 In order to preserve the Church in the purity of the faith handed on by the apostles, Christ who is the Truth willed to confer on her a share in his own infallibility. By a “supernatural sense of faith” the People of God, under the guidance of the Church’s living Magisterium, “unfailingly adheres to this faith.”
CCC 890 The mission of the Magisterium is linked to the definitive nature of the covenant established by God with his people in Christ. 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. […]
How far does this charism of infallibility extend?
CCC 2035 The supreme degree of participation in the authority of Christ is ensured by the charism of infallibility. This infallibility extends as far as does the deposit of divine Revelation; it also extends to all those elements of doctrine, including morals, without which the saving truths of the faith cannot be preserved, explained, or observed. Cf. CCC 2051
What are the organs of infallibility in the Church:
The organs of infallibility in the Church are:
a) the bishops dispersed throughout the world in union with the Pope [the Holy See];
b) ecumenical councils under the headship of the pope; and
c) the pope himself separately.
What is the deposit of Divine Revelation?
The Christian Revelation or the Sacred ‘deposit of faith‘ (the depositum fidei) [= Holy Tradition + Sacred Scripture], is that Revelation which was given in its entirety to Our Lord and His Apostles. After the death of the last of the twelve, it could receive no increment. It was, as the Church calls it, a deposit — “the faith once delivered to the saints” [Jude, 1:3 (RSVCE)] — for which the Church was to “contend” but to which she could add nothing. Cf. Catholic Encyclopedia > R > Revelation & CCC 84
Thus we pray a traditional Latin version Actus Fidei/Act of Faith with a literal English translation:
DOMINE Deus, firma fide credo et confiteor omnia et singula quae sancta ecclesia Catholica proponit, quia tu, Deus, ea omnia revelasti, qui es aeterna veritas et sapientia quae nec fallere nec falli potest. In hac fide vivere et mori statuo. Amen.
O LORD God, I firmly believe each and every truth which the holy Catholic Church teaches, because Thou, O God, Who art eternal truth and wisdom which can neither deceive nor be deceived, hast revealed them all. In this faith I stand to live and die. Amen.