There are lots of words to describe types of stories in English. Here are some of the more common ones.
Types of stories
myth = something that isn't true, but is believed by lots of people: "Don't believe in that old myth."
legend = story often about historical figures: "We studied ancient Greek legends in school."
urban legend = a modern (untrue) story where the origins are unknown: "Have you heard the urban legend about the guy who discoverd a dead mouse in his Coke and sued the company for millions?"
tale = story: "She told an incredible tale about how they first met."
fairy tale = a story with a happy ending: "Her six-year-old daughter loves listening to the classic fairy tales by Hans Christian Anderson."
old wives' tale = something which people believe to be true, but which doesn't have any scientific basis: "Saying that if you go outside with wet hair you'll get a cold is just an old wives' tale. It's not true!"
saga = a long story.
Also ongoing saga = a long story with no end: "Have you been keeping up with the ongoing saga of Sue and Bill?"
account = the facts of what happened: "He gave a good account of the meeting."
eye witness / first hand account = account by someone who was there at the time: "Eye witness reports mention that police fired the first shots".
report / newspaper report = a factual account: "Newspaper reports from the time referred to a blue BMW parked on the corner of the road."
Stories in newspapers
report = factual account
exposé = a report that uncovers the truth: "Read our damning exposé of the money for peerage scandal!"
undercover reporting = when a journalist pretends to be someone else to get to the heart of a story: "His undercover report shows the real extent of organised crime."
article = report
opinion piece = an article based on the author's opinion, rather than on the news or facts.
Type of books
fiction = non-fact: "This is a great work of fiction."
historical fiction = story about an event of person in the past.
detective story = a story about a detective: "Agatha Christie's Inspector Poirot detective stories are a great read."
murdery mystery (whodunnit) = a crime story where someone is murdered and the detective has to find the killer. "Whodunnit" is short for "Who done it?" (slightly ungrammatical English but means "Who was the killer?")
thriller = a fast-paced story: "Have you read the latest Dan Brown thriller?"
teen fiction = fiction especially aimed at teenagers.
children's fiction = stories for children.
biography = the story of someone's life, written by another person: "I'm reading George Orwell's biography."
autobiography = the story of someone written by that same person: "His autobiography is fascinating."
memoirs = the story of your past - especially written by politicians or public figures: "When her memoirs were published, there was an outcry."
romance = love story.
science fiction = fiction about aliens, or strange worlds.
short story = a complete story in a few pages.
ghost story = a story about ghosts or hauntings.
novel = a longer story.
How to tell a good story in English
Have a look at our speaking page on Telling a story in English for information on the right tenses to use, how to sequence your ideas, and how to make your story interesting.