Allegory
A story or poem with characters and events that stand for ideas about life or for a political situation. Most of the time,there is a moral to the story, or a life lesson for the reader to learn. An allegory is like a really long metaphor, which means you can find meaning beyond the literal storyline.
Analogy
An analogy compares two things that are mostly different from each other but have some aspect in common. By showing a connection between two different ideas, writers help to explain something important about one notion by using a second concept you already know about.
Fable
A short story, that often teaches a moral, typically with animals as characters. Aesop was famous for writing fables
Parable
A simple story with a moral or a story told to teach a lesson. An example of a parable is the story about a boy who cried wolf, which is used to teach us not to lie.