The theme of a story is what the author is trying to convey — in other words, the central idea of the story. Short stories often have just one theme, whereas novels usually have multiple themes. The theme of a story is woven all the way through the story, and the characters' actions, interactions, and motivations all reflect the story's theme.
But don't confuse theme with the story's plot or moral. The plot is simply what happens in the story and the order of the story's events, and the moral is the lesson that the writer wants the main character (and by extension, you) to learn from the story. Each of these serves the overall theme of the story. That is, the events of the story illustrate the theme, and the lesson that you learn relates directly to the theme.
So when you're trying to recognize the theme of a story, ask yourself what the author is trying to convey through the characters and events of the story. For instance, in The Catcher in the Rye, Holden Caulfield's actions are motivated by his not wanting to grow up, so one of the main themes of this novel is the preservation of innocence.