What sort of short story is "The Red-Headed League" by Sir Arthur Conan Doyle?
The short story "The Red-Headed League" by Sir Arthur Conan Doyle is in the subgenre of mystery story. Mystery stories are a subgenre because they have particular and definitive elements and features. For instance, a mystery must solve a mystery, must have a detective (or a homegrown sleuth like Nancy Drew), must have a foul crime, must answer Who did it? How was it done? and Why was it done? Major genres don't have such particularly defining features and are built around character development as opposed to attaining some end such as the solution of a mystery.
"The Red-Headed League" was first published in 1891 during the reign of Queen Victoria of England and is therefore a Victorian novel. Arthur Conan Doyle was eventual knighted by King Edward VII for his contribution as a doctor in the Boer War, which is when he became Sir Arthur Conan Doyle. Born in Scotland, Doyle trained in medicine under Dr. Joseph Bell who had legendary powers of observation and upon whom Doyle later built the basis of the character of Sherlock Holmes. Doyle was influenced by Edgar Allen Poe in his development of the Sherlock Holmes stories, the first one of which he introduced in 1887, that being A Study in Scarlet. All these elements together describe what sort of story "The Red-Headed League" is.