The End Game is the term NetHack uses to refer to the dungeon branch containing the Elemental Planes and the Astral Plane. The branch is internally referred to as the Elemental Planes, although the Astral Plane is considered separate from them.
To enter the End Game, you must take the up staircase on the the first floor of the dungeon or else quaff a cursed potion of gain level while holding 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.