The office of teacher.
In theology, it refers to the teaching office of the Church.
This office was communicated to the Church formally by Christ, when He said: "Going, therefore teach ye all nations" (Matthew 28).
The Church exercises this teaching power infallibly in matters of faith and morals, in virtue of the promise of Divine assistance given her by Christ, "And behold I am with you all days even to the consummation of the world" (Matthew 28).
This command to teach and this promise of special assistance were given to the Apostles only.
Therefore the actual holders of the teaching office in the Church are the pope and the bishops, as the successors of Saint Peter and the other Apostles.
The pope and the bishops constitute the magisterium of the Church, or the Ecclesia Docens (the Church Teaching).
The prerogative of teaching is exercised ordinarily, i.e., in the ordinary daily teaching of the Church, by the individual bishops.
This goes on uninterruptedly throughout the years.
The individual bishop as such (the Bishop of Rome, successor of Saint Peter, excepted) is not infallible.
When all the bishops in the world agree in their teaching on a particular doctrine of faith or morals, which is not solemnly defined, this constitutes an infallible teaching of the Ecclesia Docens, because the Church as a whole cannot fall into error in these matters.
The pope enjoys the prerogative of infallibility in his official capacity as successor of Saint Peter, and hence Supreme Pastor of the Church.
When the pope solemnly defines a truth to be de fide for the whole Church this is called a solemn exercise of the magisterium.
The pope may exercise this solemn prerogative either in conjunction with an aecumenical council, or on his own authority.
This magisterial power of the Church Teaching involves a corresponding obligation on the part of the Church Hearing (Ecclesia Discens): "He that heareth you, heareth Me" (Luke 10).
The faithful are obliged in virtue of the infallible teaching power (magisterium) of the Church Teaching, whether this power is exercised ordinarily or solemnly, to submit their understanding to the teaching of the Church.
This is called the assent of faith.
New Catholic Dictionary