A grotesque character is someone who invokes both empathy and disgust. Characters such as Gollum from Tolkien's "Lord of the Rings" or the Quasimodo from "The Hunchback of Notre Dame" are examples of grotesque characters.
Dr Frankenstein's creature and the Beast from "Beauty and the Beast" also fit into the mold of a grotesque. Modern characters, such as Sheldon from "The Big Bang Theory," could be considered grotesque. Physical disfigurements or mental deficiency is not a requirement for a grotesque; crippling social ineptitude can be the defining trait of a grotesque. The term "grotesque" may also define a situation or environment that is twisted, odd or unbelievable, and often hints at an impending, unavoidable change.