The magical creature that transforms into a person's worst fear is a Boggart. In the Harry Potter world, a Boggart is a shape-shifter whose true appearance is unknown because it changes form when confronted by a person who fears it.
Boggarts are introduced in Defence Against the Dark Arts lessons in Harry Potter and the Prisoner of Azkaban. Professor Lupin teaches students to use the Riddikulus charm, which forces the Boggart into a humorous or ridiculous form and weakens its frightening effect. Neville Longbottom's Boggart appears as Professor Snape, while Harry's would likely take the form of a Dementor.
Boggarts are often confused with Dementors because Dementors represent fear and produce a similar atmosphere of terror, but they are different creatures. Grindylows are aquatic beings, and a Lethifold is a separate, highly dangerous dark creature. The idea also draws on the English folklore boggart, although Rowling's version uniquely changes shape according to the observer's deepest fear.