Exorcism
In Catholic Christianity, exorcisms are performed in the name of Jesus Christ. A distinction is made between a formal exorcism, which can only be conducted by a priest during a baptism or with the permission of a bishop, and "prayers of deliverance" which can be said by anyone.
The Catholic rite for a formal exorcism, called a "Major Exorcism", is given in Section 11 of the Rituale Romanum. The Ritual lists guidelines for conducting an exorcism, and for determining when a formal exorcism is required. Priests are instructed to carefully determine that the nature of the affliction is not actually a psychological or physical illness before proceeding.
Blue Exorcist
In contrast, there are those called Exorcists, people who train to destroy demons who act in a damaging manner in Assiah. With more than two thousand years of existence, this group has several branches all over the world, being secretly under the command of the Vatican itself and acting in a subterfuge way to the masses.