"The Magisterium is aive with speculation"

Based on the Magisterium in our world, this is the organisation of the church. In our own world it is the "teaching authority of the Roman Catholic Church", and more information can be found on the Wikipedia.

The Magisterium in Lyra's world is far more powerful and has control over scientific research and education facilities. It is made up of several different sub-organisations that vie for power, the strongest leading the authority with their own particular doctrine. The two primary sub-organisations that have control are the Consistorial Court of Discipline and the Society of the Work of the Holy Spirit.