What is a Short Story?

A short story can be defined in only the most general terms, and for every "rule" named here, we can find short stories that ignore the rule. On the other hand, many stories fail to interest readers, and the reason they fail is most often because they ignore one or more of these general rules.

Douglas Glover, in The Enamoured Knight, writes, The Greeks called their novels tales of suffering for love. If they weren't about suffering for love, they wouldn't be tales. A story consists of someone wanting something and having trouble getting it. There are no stories about people who start out happy and contented, remain happy and contented throughout, and end up happy and contented.

Writing a short story is part craft, part intuition. At its most basic, a story has a beginning, a middle, and an end, each with it's own specific purpose. Just as the beginning of a novel introduces characters, setting, and goals, the beginning of even a short short story also sets the stage and introduces the main character's problem or goal. The middle introduces and deepens complications or conflict, and the end forces the POV character to make a decision that resolves his or her problem, goal, or need, so that the character succeeds or fails and readers, knowing the outcome, feel satisfied.

Characteristics of a Short Story

captures a thought provoking or profound moment or incident on paper and resolves the problem inherent in that moment in such a way that a change occurs for the character, and possibly for the reader. The change that occurs in a character may be subtle, such as a shift in attitude, or it may be overt, such as a change in fortune.
has a definitive beginning, middle, and end. It begins as close to the end as possible, works through some complications to arrive at a climactic moment, and then ends.
usually involves a sympathetic character that overcomes some hurdle to achieve a meaningful goal.
wastes no time getting started but plunges into the moment in time to be explored.
focuses on one central character that interacts with as few other characters as possible around a single incident in a single setting.
focuses on only those details that elaborate on and help to resolve the story problem.

Another description

A short story is a narrative involving a conflict between two poles (A vs. B). This conflict needs to develop through a series of actions in which A and B get together again and again and again (three is a good number to start with, but there can be more).

Finally, one more definition for writing short fiction:

By a story I mean a narrative that extends through a set of articulations, events or event sequences, in which the central conflict is embodied once, and again, and again (three is the critical number here looking back at the structure of folk tales) such that in these successive revisitings we are drawn deeper into the soul or moral structure of the story.
~ Douglas Glover (The New Quarterly, #87, Summer 2003)