Supervised by Lord El-Melloi II, a classroom where Modern Magecraft is taught by three to five professors (fluctuating over time).
For a time the El-Melloi Classroom was the only place teaching Modern Magecraft, so there are some who use the terms interchangeably. Strictly speaking, however, the El-Melloi Classroom is only a single part of the Faculty of Modern Magecraft, a small seminar of about fifteen students. Including auditing students, that number jumps to about fifty. Left alone, there would be a flood of students joining officially, but there may or may not exist a self-governing body of students within the classroom keeping strict control of it. And there may or may not be a black market for tickets to gain entry.
In one memory scene, Waver borrows money from Melvin in order to purchase the classroom. Really what he is purchasing is "the right of have a classroom in the headquarters of the Clock Tower." A classroom in the Clock Tower is something like a castle, so at the time Waver had to borrow enough money to buy a castle.
Furthermore, during the course of the events of Case Files, the number of teachers and students in the Faculty of Modern Magecraft grew considerably, reaching the numbers they have today.