Trap Door is an enemy from Final Fantasy IV: The After Years. The only way they can be fought is by pressing the action button on a locked door in the Sealed Cave or Rydia's Challenge Dungeon. They can never be fought as a random encounter.
Rydia can usually kill Trap Doors with one cast of Bio depending on the Moon Phase. This is the fastest way to kill Trap Doors which is important when repeatedly going through Rydia's Challenge Dungeon.
If Rydia has not learned Bio yet, simply have her cast Stop on the Trap Door as soon as she can. This will easily prevent the Trap Door from casting Ninth Dimension which can instantly kill someone and will also prevent the Trap Door from summoning a monster when it is low on HP.