Is the Pope really infallible?

Yes. But the Pope speaks infallibly only when he defines a dogma in a solemn ecclesiastical act (“ex cathedra”), in other words, makes an authoritative decision in doctrinal questions of faith and morals. Magisterial decisions of the college of bishops in communion with the Pope also possess an infallible character, for example, decisions of an ecumenical council.

The infallibility of the Pope has nothing to do with his moral integrity or his intelligence. What is infallible is actually the Church, for Jesus promised her the Holy Spirit, who keeps her in the truth and leads her ever deeper into it. When a truth of the faith that has been taken for granted is suddenly denied or misinterpreted, the Church must have one final voice that authoritatively says what is true and what is false. This is the voice of the Pope. As the successor of Peter and the first among the bishops, he has the authority to formulate the disputed truth according to the Church’s Tradition of faith in such a way that it is presented to the faithful for all times as something “to be believed with certainty”. We say then that the Pope defines a dogma. Therefore such a dogma can never contain something substantially “new”. Very rarely is a dogma defined. The last time was in 1950.

“The college or body of bishops has no authority unless united with the Roman Pontiff, Peter’s successor, as its head.” As such, this college has “supreme and full authority over the universal Church; but this power cannot be exercised without the agreement of the Roman Pontiff.” (CCC 883)

“”The individual bishops are the visible source and foundation of unity in their own particular Churches.” As such, they “exercise their pastoral office over the portion of the People of God assigned to them,” assisted by priests and deacons. But, as a member of the episcopal college, each bishop shares in the concern for all the Churches. The bishops exercise this care first “by ruling well their own Churches as portions of the universal Church,” and so contributing “to the welfare of the whole Mystical Body, which, from another point of view, is a corporate body of Churches.” They extend it especially to the poor, to those persecuted for the faith, as well as to missionaries who are working throughout the world.” (CCC 886)
“Bishops, with priests as co-workers, have as their first task “to preach the Gospel of God to all men,” in keeping with the Lord’s command. They are “heralds of faith, who draw new disciples to Christ; they are authentic teachers” of the apostolic faith “endowed with the authority of Christ.”” (CCC 888)
“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 889)
“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. The exercise of this charism takes several forms:” (CCC 890)
“”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.” (CCC 891)
“Divine assistance is also given to the successors of the apostles, teaching in communion with the successor of Peter, and, in a particular way, to the bishop of Rome, pastor of the whole Church, when, without arriving at an infallible definition and without pronouncing in a “definitive manner,” they propose in the exercise of the ordinary Magisterium a teaching that leads to better understanding of Revelation in matters of faith and morals. To this ordinary teaching the faithful “are to adhere to it with religious assent” which, though distinct from the assent of faith, is nonetheless an extension of it.” (CCC 892)