What are the similarities/differences between allegory, fable, and satire?
Each of these performs a different function..
In an allegory, each major character represents an idea or philosophy or a segment of society. Gilligan's Island had the working class (Gilligan and the Skipper), high society (Mr and Mrs Howell), academia (the Professor), homespun agrarian middle America (Mary Ann) and the glamorous entertainment industry (Ginger). This was not a coincidence, it's how the show was designed. The Breakfast Club did something similar, representing the major cliques of most American schools.
A fable is a short story, often with animals instead of people, that illustrates a moral point. It always ends with "And the moral of the story is..." "The Ant and the Grasshopper," "The Fox and the Grapes," and all the rest of Aesop's Fables are the most famous examples of this.
Satire is a type of story where human behavior is held up for ridicule. There is gentle Horatian satire ("Oh look, the king picks his nose and moves his lips when he reads, just like us commoners!") or vicious Juvenalian satire ("The wealthy would work us all to death for a quick profit, and then drink our blood for fun!").
Allegory tends to be book- or feature-length. Fables tend to be shorter, maybe three pages at the most. Satire can work at any length and in any idiom; it tends to be humorous, but this is far from a necessary defining feature. Also, a story can have elements of more than one of these. Tom Sawyer was allegorical and satirical, for instance.