A fable is a tale that teaches a lesson. Fables are meant to entertain but also tell important truths about the way people act.
In almost all fables, the characters are animals who are anthropomorphised (given human characteristics such as speech). Fables might state a lesson or moral at the end of the story.
Some familiar sayings that originated from fables include:
Little is known about the ancient Greek writer Aesop (c. 620 B.C.E.–c. 560 B.C.E.), whose stories of clever animals and foolish humans are considered Western civilization's first morality tales. He was said to have been a slave who earned his freedom through his storytelling and went on to serve as advisor to a king. Both his name and the animist tone of his tales have led some scholars to believe he may have been Ethiopian in origin.
Anthropomorphism, or animals with human capabilities, is the common thread throughout Aesop's fables. The most famous among them are "The Tortoise and the Hare," in which the plodding turtle and the energetic rabbit hold a race. The arrogant hare is so confident that he rests and falls asleep halfway; the wiser tortoise plods past and wins. "Slow but steady wins the race," the fable concludes.
Several phrases are traced back to the fables of Aesop, such as "don't count your chickens before they are hatched," which concludes the tale of the greedy "Milkmaid and Her Pail." In "The Fox and the Grapes," a fox ambles through the forest and spies a bunch of grapes. Thirsty, he tries in vain to reach them but finally gives up and walks off muttering that they were likely sour anyway. From this comes the term "sour grapes."
(Adapted from: Encyclopedia of World Biography 2005 http://www.encyclopedia.com/topic/Aesop.aspx#1)