The Royal Quarters is a new dungeon area added in Patch 2.4. It extends off to the right from the staircase in Leoric's Manor, past rubble that was previously blocking that passageway. This is a non-random area, with the same layout each time.
Set Dungeon Tomes
This area is notable for holding the Tome of Set Dungeons, a lore book found in the same location each game. It will drop a clue to find the location of the Set Dungeon corresponding to the item set your character is wearing.