A nutcase is a very informal, often unkind word for someone who behaves in a wild or crazy way. It can be a joke between friends (“You’re a nutcase for swimming in that cold river”) or a real insult. It is not a medical word, and careful people avoid it when talking about real mental illness. Nut here is old slang for the head, and then for a person whose head seems “off”; case is like “a type of person,” the same idea as headcase. The word belongs in spoken, casual English, not in essays or polite talk with strangers.