Please identify the characteristics of the short story, such as seen in "The Gift of the Magi."

Edgar Allen Poe used Aristotle's theory of unity for a play as also the criteria for a good short story. A short story should have unity of time, unity of place, and unity of action. This means that the plot will usually be fixed in one place, with the action happening over a relatively brief period of time (the setting). The story line will not be elaborated into complicated plots and subplots but will usually deal with one conflict, then focus on its resolution. Of course there are exceptions, but this is a good rule of thumb to follow.

You can see these guidelines applied in "The Gift of the Magi." The reader only sees things from Della's point of view although what happened to Jim when he made his purchase is just as important to the outcome of the story. 

An example of exception is the Nebula award winner "They're Made Out of Meat" by Terry Bisson.  The whole thing is just a dialogue between two extraterrestial beings.  There is no setting, no action, just talk; but it's great!