what is a "fable"?

In its strict sense a fable is a short story or folk tale embodying a moral, which may be expressed explicitly at the end as a maxim. "Fable" comes from Latin fabula (meaning 'conversation', 'narrative', 'tale') and shares a root with faber, "maker, artificer." Thus, though a fable may be conversational in tone, the understanding from the outset is that it is an invention, a fiction. A fable may be set in verse, though it is usually prose. In its pejorative sense, a fable is a deliberately invented or falsified account.
A fable often, but not necessarily, makes metaphorical use of an animal as its central character. Medieval French fabliaux might feature Reynard the fox, a trickster figure, and offer a subtext that was mildly subversive of the feudal order of society. A familiar theme in Slavic fables is an encounter between a wily peasant and the Devil. But the device of personification may be extended to anything inanimate, such as trees, flowers, stones, streams and winds.
In some usage, "fable" has been extended to include stories with mythical or legendary elements. The word "fabulous" strictly means "pertaining to fables," although in recent decades its metaphorical meanings have been taken to be literal meanings, i.e. "legendary," "mythical," "exaggerated," "incredible." An author of fables is called a fabulist.
Some modern fabulists

George Ade, "Fables in Slang" and other titles
Don Marquis, author of the fables of archy and mehitabel
James Thurber (1894-1961), Fables For Our Time.
Damon Runyon
Sholem Aleichem
Bill Willingham author of Fables graphic novels
Ambrose Bierce
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Notable fables

Panchatantra
Hitopadesha
Baital Pachisi
Stone Soup
The Little Engine that Could
Jonathan Livingston Seagull
Watership Down
The Lion King
Emperor's New Clothes (fable)
Fables and Parables by Ignacy Krasicki
The Fox and the Cock by James Thurber
Animal Farm by George Orwell
The Boy Who Cried Wolf