is an NPC that you will encounter in several places as you progress through the game. It's not required that you find him in all of his locations or talk to him in order for him to progress.
You will meet him before entering the Teacher's Archives and he will help you to defeat Uumu inside by puncturing the boss' protective membrane, allowing you to deal damage to it. He will then appear next to a large tank and make it possible for you to use the Dream Nail on Monomon the Teacher.
After this event, Quirrel will move to his final location in Blue Lake:
Interact with him here and the achievement will unlock.