End Game is NetHack's displayed term used to refer to the dungeon branch containing the Elemental Planes and the Astral Plane. The branch is internally referred to as the Elemental Planes, even though the Astral Plane is not an elemental plane.
To enter the End Game you must go above the first floor of the dungeon (by climbing the stairs or drinking a cursed potion of gain level) with the Amulet of Yendor. Once you enter this branch, you cannot return to the Dungeons of Doom. All of the planes are no-teleport and have undiggable floors, so it is impossible to move backwards to an earlier plane or to return to any of the earlier dungeon branches.