@Kyujyu "Uh... Magic + Auditorium = Magitorium?"
Not quite.
"tor" is a root which turns words into personal agent nouns. As an example of that, consider ianua [door/entryway] + tor = ianuator [Doorkeeper].
Ianuator is the originating term which would eventually shape into the rather differently applied word "Janitor".
As a more straightforward example, Gladius+tor = Gladiator [Someone who wields a (Gladius) sword].
"ium" has a
variety of applications, but is typically seen as a suffix which designates a location dedicated to the preceeding phrase.
As examples, consider Natatorium and Sanatorium:
Nare- "to swim" -> Natator "swimmer" -> Natator+ium = ("place for swimmers") -> Natatorium: [Indoor] swimming pool
Sanat- "to heal" -> Sanitorius "health-giving" -> Sanator+ium = ("health-giving place") -> Sanatorium: An establishment dedicated to long-term medical care
Incidentally, Auditorium originates from the same principle, with "Auditorius" meaning "of or for hearing", indicating that an Auditorium is a place dedicated to listening [to things].
Magi (Also Latin; plural of Magus "a magician") + tor + ium gets a bit weird, with trying to determine the interaction of "tor" with "magi", since "magi" is already a noun.
However, we can look to Senator for an existing example of a similar word. Senate "a chamber of elders" + tor = Senator "An individual from the Senate"
Hence, we can conclude that Magitorium would mean: "A place for individuals from the Magi".
Based on typical progression of meaning for such terms, we can form that interpretation up a bit further, giving us a final meaning of: "A gathering place for Magi".