There are two meanings for “exorcist”…
(1) a minor orders ordination, comparable to taper-bearer (aka candlebearer), lector, acolyte, or cantor. All minor orders were suppressed in the mid 1960’s within the Roman Rite; the fullness of their ordination is included in the diaconal ordination even if not separately imposed.
(2) a priest (or very rarely deacon) appointed with a specific faculty to investigate, evaluate, and treat demonic possession, and/or to perform the ritual of the greater exorcism, to expel demons.
The lesser exorcism, also known as the baptismal exorcism, is used at every priest-performed baptism…
The greater exorcism is used to combat possession, and is not a default faculty in any church to my knowledge. Rome’s always been a bit twitchy about who is permitted to investigate demonic possessions, as the mentally ill are often confused for the possessed (and now, vice versa).
Every Deacon and Priest has the power to banish demons by commanding them out, and in those Churches Sui Iuris where exorcist is still used, every ordained exorcist and subdeacon as well. Not every one of them has permission to do the investigations and rituals to deal with possession.
newadvent.org/cathen/05711a.htm discusses the pre-vatican II practice, and the history of the order of exorcists.