For children! ...Sort of?

Ramon Llull, a Catalan mystic and poet from the 13th century, is one of the earliest known authors to use "movables" in his books.  His work was intellectual and aimed at other scholars, but in the 18th century, these "movables" were beginning to be produced for children's amusement thanks largely to London publisher Robert Sayer.

As you see here, pop-up books share similar physical qualities to the board books aimed at toddlers.  

David A. Carter's One Red Dot alongside one of the popular Osborne's Touchy-Feely books 

 Square proportions, and both using durable cardboard covers.  The cover protects the delicate pieces of the pop-up and provides the counterweight for the constructions once the books is open.

Pop-up books also share traits with children's picture books.  Both rely heavily on illustration and are typically light on story, perhaps a paragraph or two per page.  Here's a side-by-side comparison of the art-to-word ratio:

Sabuda, Robert.  (2005)  Winter's Tale.  New York, New York: Little Simon.
Van Allsburg, Chris.  (1985)  The Polar Express.  New York, New York: Houghton Mifflin Company.

Otherwise, the pop-up stands out among all other book forms.  Whereas, most other books are two-dimensional, these rock the third.  The Wild Things didn't jump out at you, but the monsters in Maurice Sendak's Mommy? will!

Sendak, Maurice, Arthur Yorinks & Matthew Reinhart.  (2006)  Mommy?  China: Michael di Capua Books.

However, as the art and creativity behind pop-up books grew, so did its audience.  Today the book form has fans among both children and adults.  It is not difficult to trace that shift in consumption as the constructions became more delicate and elaborate.

References
http://www.library.unt.edu/rarebooks/exhibits/popup2/introduction.htm