There are many elements that are important for a short story. In view of this, I will have to be selective. I will mention: setting, characterization, and plot.
The setting of a short story sets the tone. This is important for many reason, because certain settings create certain moods. This can help the reader to expect things that are appropriate to the setting. Or if the author introduces atypical elements into the setting, then his tension can create interesting results.
Characterization is important, because to make a story enjoyable there has to robust character development. In addition, the more the reader appreciates the character, the more powerful the story becomes. In fact, character development is one of the most important elements of a successful short story.
Finally, the plot is what actually happens in the story. There should be a conflict, rising tension, and resolution.
Setting: provides context for the action of the story and gives the reader important background information. For example, a in story about dating set in Saudi Arabia where public displays of affection are illegal, a kiss at the front door at the end of a date has a significance it wouldn't in a story set in the US. Often setting (e.g. a blasted heath) creates mood and atmosphere as well.
Characterization: We often read stories because we are interested in what happens to characters. The better constructed the characters, the more interesting and engaging the story.
Plot: Action is what drives the story and makes us want to continue reading. We want to find out what happens next. Without action, in general, there is no story.
Dialogue: A short story can be told entirely through exposition, but dialogue lets the characters speak in their own voices, adding depth to characterization.