One frightening story is Neil Gaiman's brutal retelling of Snow White from the Queen's point of view in "Snow, Glass, Apples." Joyce Carol Oates's "Where Are You Going, Where Have You Been?" is a story with a more modern feel, yet it has supernatural undertones and real menace in it.
"The Monkey's Paw" by W. W. Jacobs is one story not-to-be-missed. It concerns bad luck, bad decisions and one final, terrible choice.
Edgar Allen Poe is one of the classic names in short horror fiction and "The Tell-Tale Heart" is one of his most famous work. One of the things that makes it so good is the psychological portrayal of the narrator's descent into madness, but with a believable supernatural element that keeps the reader guessing.
Another great name in horror belongs to H. P. Lovecraft, master of pulp fiction and the never-quite-glimpsed. His novella, The Dunwich Horror, is a masterpiece of the genre, with characters losing their sanity in a struggle against mysterious and unearthly creatures.
Another story in the not-to-be-missed category is Shirley Jackson's, "The Lottery." Although it has none of the usual horror story tropes, its stark portrayal of human greed and wickedness, followed by the ultimate sympathetic turnaround, make this tale one that never stops haunting.