The hooded figure is a non-player character in the first Final Fantasy. They have no officially given name. They appear in Cornelia after defeating Garland at the Chaos Shrine, but talking to them yields silence and they walk out of the town.
After the Four Fiends are defeated, speaking to them at Cornelia will direct the party to the Chaos Shrine, where the figure reveals the entrance to the Labyrinth of Time. They inform the Warriors of Light about the nature and rules of the Labyrinth and leave it available for them to challenge.
During the warriors' progression through the labyrinth, the hooded figure may appear at the start of any floor to provide help in the form of additional time, items, removing random encounters for the floor, or a possible way to leave.
The hooded figure may also appear in a level's puzzle, acting as the person the Warriors of Light report to upon completion of their task. They oversee it is done correctly, and allow passage to the next floor. Alternatively, if the warriors failed, they reset the floor and force them to re-do the puzzle.
When the Warriors of Light reach the Space in Time the hooded figure leads them to the Lord of Time, Chronodia, before disappearing. Chronodia reveals that the Labyrinth of Time is an illusion, created for the purpose of luring powerful warriors and testing their mental and physical fortitude. If successful, they face Chronodia and seek to claim her power for their own, just as Chronodia seeks to add their power into hers. After the battle, the hooded figure will remain just outside the labyrinth as before.
It is unknown if the hooded figure is an illusion created by Chronodia, a form of Chronodia herself, or a wholly separate entity who acts as her proxy.