Is The Little Prince an allegory, fable, fairy tale, or children's book?

I think that you could put The Little Prince into any of those categories. Let's examine each one:

An allegory is a symbolic story in which elements of the story such as characters or setting are meant to symbolically represent certain concepts. The Little Prince has often be thought of as a Christian allegory, with the Prince representing Christ, the snake representing the serpent from the Garden of Eden, and so forth. You can read...

I think that you could put The Little Prince into any of those categories. Let's examine each one:

An allegory is a symbolic story in which elements of the story such as characters or setting are meant to symbolically represent certain concepts. The Little Prince has often be thought of as a Christian allegory, with the Prince representing Christ, the snake representing the serpent from the Garden of Eden, and so forth. You can read more about this connection in 

A fable is a short story or tale that is meant to teach an explicit lesson or moral. The Little Prince can be seen to work in that way -- one point of the story is to explain the nature of love, which is expressed explicitly in the story by the Fox's secret, about how the essential is invisible to the eye.

A fairy tale is best understood in this context as a story in which magic or magical creatures figure in the plot. A fairy tale is different from a fable in that it can contain hidden meaning, or require interpretation to understand, whereas a fable's moral is clear. Again, The Little Prince can be understood this way: there are magical creatures, and even the imagination, in the form of the drawing of the sheep, can be seen as "magical;" the Prince's tiny asteroid is not unlike a fairy tale kingdom; the ultimate meaning of the book is not clear at all, and, as any quick search for "the meaning of The Little Prince" will show, very much open to interpretation.

A children's book is more a marketing category than a literary genre, but I think it is fair to say that most people would consider The Little Prince to have a broad appeal to children. Interestingly, this distinction between books for children and grownups is a theme of The Little Prince, in that grownups tend to forget or overlook the important things children can see clearly. This might explain the form of the book, which arguably tries to articulate deep truths in a way a child might express them.

Ultimately, I think that The Little Prince is a book that defies easy categorization. That it could be any of these genres implies that perhaps it  belongs in none of them. What is useful is paying attention to how your understanding of the book changes when you read it using any of these genre conventions as a guide!