The invisible woman of Cornelia (previously known as the invisible man of Cornelia) is a non-player character in the original release of Final Fantasy whose sprite does not appear due to a bug. The invisible woman is in Cornelia and has dialogue, and the player can still talk to her, resulting in an awkward moment where the player seems to be talking to no one.
The bug is triggered as a result of a mistake in the game's programming. Whenever the player enters a room, the characters who remain outside of it become invisible to simulate the player's inability to see outside the room, while the characters inside the room would in turn become visible. The invisible woman's sprite is programmed to be visible inside rooms, but she is placed outside of a room. Since there is no room near her sprite, there is no way to ever actually make her visible without cheating.
The invisible woman can be seen by using a Game Genie (or other similar device) to enable a walk-through-walls code, so that the player may walk in the area outside of a room while sprite visibility is set to the configuration while inside of a room.
Before her discovery via the Game Genie, the Nintendo Power January 1991 issue documented this bug as the "Invisible Man" in its Classified Information section, which documented bugs, cheats, and other oddities submitted by readers.
Most subsequent versions of Final Fantasy "fixed" the invisible woman by outright removing her, though interestingly, one of her two pieces of dialogue has been retained, and given to another female NPC in Cornelia Castle (who previously had generic dialogue used by other NPCs). This solution may have something to do with the NPC's original purpose. In the NES version, she is programmed to say "The Princess was looking for you!" unless the princess has been rescued, and the player has the Lute. This message serves as a clue to the player to talk to Princess Sarah, who then gives the Warriors of Light the Lute (which is required to proceed past a certain point in the game).
However, from the WonderSwan Color version onward, receiving the Lute from the princess was made a mandatory scripted event, making it impossible for the player to miss this item. As such, the invisible woman's first message was no longer necessary and her second message could be consolidated into another NPC with otherwise redundant dialogue.