The word fable comes from the Latin fabula, where it simply means "story."
In the English-speaking world, the word is closely associated with Æsop's Fables, a collection of stories attributed to the ancient Greek "Æsop the Slave," who lived in the sixth century BC. In 1483, William Caxton, one of the first printers in England, published an English translation. Æsop's fables are short stories, obviously fictitious, usually involving talking animals as characters, which make some point about morality or human nature. The point is sometimes stated explicitly in a final sentence, sometimes labelled as the "moral:" "Some men underrate their best blessings," or "There is no believing a liar, even when he speaks the truth."
An example of one of Æsop's fables
Two travellers, worn out by the heat of the summer's sun, laid themselves down at noon under the widespreading branches of a Plane-Tree. As they rested under its shade, one of the Travelers said to the other, "What a singularly useless tree is the Plane! It bears no fruit, and is not of the least service to man." The Plane-Tree, interrupting him, said, "You ungrateful fellows! Do you, while receiving benefits from me and resting under my shade, dare to describe me as useless, and unprofitable?'
Some men underrate their best blessings. 
Notes and references
- Aesop's Fables, translated by George Fyler Townsend Project Gutenberg eText