If you are looking for something for younger folders - something less complicated than the models found in the books mentioned above, there are many simple origami models that can be used to help spark mathematical thinking and exploration with elementary school students. One example is the jumping frog (instructions available on Origami USA's diagram page).
Unlike most traditional origami, the frog is made from a rectangular sheet of card (3x5 works well) - the frog is best made from card stock to give it a stiffness that helps it hop. The frog can be colored after it is completed to make it look more frog-like (younger kids particularly like this - and the jumping).
"Dissecting" the frog by opening it up after it is folded reveals a crease pattern that can spark many mathematical conversations.
What I like about this model is that every fold involves the constuction of an angle bisector - every crease cuts an angle in half (OK, some of the bisectors are cutting a 180 degree angle in half, but I'm counting those too). This makes it easy to talk about the resulting angles and shapes. Can you identify all the angles? Can you name all the shapes? Can you think of other mathematical conversations you could have with young students about the model and its crease pattern?