What is the Magesterium?

Is it the church or a group of bishops or a book?

Here’s an answer from one of the CAF apologists: forums.catholic.com/showthread.php?t=53316

Here is how the Catechism defines it:

85 "The task of giving an authentic interpretation of the Word of God, whether in its written form or in the form of Tradition, has been entrusted to the living teaching office of the Church alone. Its authority in this matter is exercised in the name of Jesus Christ."47 This means that the task of interpretation has been entrusted to the bishops in communion with the successor of Peter, the Bishop of Rome.

86 "Yet this Magisterium is not superior to the Word of God, but is its servant. It teaches only what has been handed on to it. At the divine command and with the help of the Holy Spirit, it listens to this devotedly, guards it with dedication and expounds it faithfully. All that it proposes for belief as being divinely revealed is drawn from this single deposit of faith."48

87 Mindful of Christ’s words to his apostles: “He who hears you, hears me”,49 the faithful receive with docility the teachings and directives that their pastors give them in different forms.

You can read more at vatican.va/archive/ccc_css/archive/catechism/p1s1c2a2.htm

none of the above

It is quite simply the “teaching authority of the church”

It is held by the college of bishops, including the Pope.

As has been mentioned, bishops have the office of teaching, sanctifying, and governing. Magisterium refers to the teaching part. More from the CCC:

The teaching office

888 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.415 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."416

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."417

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. The exercise of this charism takes several forms:

891 “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.418 When the Church through its supreme Magisterium proposes a doctrine "for belief as being divinely revealed,"419 and as the teaching of Christ, the definitions "must be adhered to with the obedience of faith."420 This infallibility extends as far as the deposit of divine Revelation itself.421

892 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"422 which, though distinct from the assent of faith, is nonetheless an extension of it.

vatican.va/archive/ccc_css/archive/catechism/p123a9p4.htm

(follow the link for the whole section, which includes discussion on the sanctifying and governing aspects)

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