A story's deeper meaning is called it's theme.
When you read/watch/hear a story, the underlining message you get from the story is a 'theme' you see, usually represented by some type of symbolism. Some stories even have multiple themes, and themes usually fall into two categories - what the reader 'thinks the story is about,' and it's thematic statement being 'what the /work/ says about the subject.'
Hope this helped :-)
