The Magisterium is not a group of people at all. The Magisterium is the authority that the Catholic Church claims, derived from the authority given to the apostles, to interpret Scripture and Sacred Tradition so as to maintain the true faith:
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.
(Dei Verbum, the Second Vatican Council's Dogmatic Constitution on Divine Revelation, chapter 10, section 2; quoted in Catechism of the Catholic Church, paragraph 85)
This teaching authority is vested in the hierarchy of the Church; paragraph 85 continues:
This means that the task of interpretation has been entrusted to the bishops in communion with the successor of Peter, the Bishop of Rome.
In other words, the Pope, and the bishops who are in communion with him (that is, who recognize him as the supreme head of the Church on Earth and who accept his doctrinal statements) together act (at the prompting of the Holy Spirit) to explain to the People of God what the revealed truths of the Christian faith are, by interpreting Sacred Scripture and Sacred Tradition. When the Pope, or the bishops acting in an ecumenical (church-wide) council led by the Pope, declare that something is dogma, they do so in virtue of the Magisterium which they exercise, inherited from the apostles.