The Dungeon is a maze that's beneath the school. Students with permission can go there, in parties of up to 3.
The Protagonist is always brought, along with up to 2 members of the players choice.
If you've done the Submissive plotline, you can only choose one person to bring with you, as Cassandra will permanently be in your second slot unless you decide to break out of that relationship with her.
There's currently 13 floors, three Sigils and five Secret Notes.