Short stories are a type of fiction, usually between 1000 and 4000 words long. Shorter than this and we have Flash Fiction, longer and we have a Novella. Whilst the 4000 words long limit is generally accepted, the Science Fiction and Fantasy Writers of America allow up to 7500 in their competitions.
Short stories began in the times where writing down a tale was not usual or even possible. When stories were related using the oral tradition they had to be short enough for the teller to remember.
Interest in short stories has grown and waned over time and current interest from publishers is low. The heyday of the printed short story was probably 1850-1950 when magazine and periodical sale were at their highest. Many famous writers from Charles Dickens to John Steinbeck found their initial readership through short stories published in magazines.
The classic definition of a short story is one that can be read in a single sitting. This was certainly Edgar Allen Poe’s view. HG Wells agreed saying “ [short story writing is] The jolly art, of making something very bright and moving; it may be horrible or pathetic or funny or profoundly illuminating, having only this essential, that it should take from fifteen to fifty minutes to read aloud."
Somerset Maugham said that “[a short story] must have a definite design, which includes a point of departure, a climax and a point of test; in other words, it must have a plot". Chekhov disagreed, in his view “a short story should have neither a beginning nor an end. It should just be a "slice of life", presented suggestively”
For more information on the history of The Short Story see this Wikipedia article