What is the physics behind origami? If we fold a paper and then apply pressure on the newly formed crease, it seems that the paper's surface gets a permanent deformation but what exactly has happened to the paper at a molecular scale?
 A: Basically, a fold or crease in paper will remain because the structure of the fibers in the paper have become irreversibly damaged. This happens because the paper is bent/compressed beyond its elastic limit.
Chemically, paper is mainly composed of cellulose  from plant fibers. Cellulose is an organic polymer, which has D-glucose units connected through hydrogen bonds. These bonds form between the oxygen atom of the one-hydroxyl group belonging to the glucose and the hydrogen atom of the next glucose unit. These are microscopic properties of paper, but to understand what happens when we fold paper or do Origami, it is sufficient to learn what is happening macroscopically.
All materials have what is called an elastic limit and a plastic region. The elastic limit is the point at which a material will bend but still return to its original position without any permanent change or damage to its structure. Further deforming the material beyond this limit takes it to its plastic region. At this point any structural or physical changes become permanent and the paper will not return to its original form.
Every material has a different elastic limit or yield, and plastic region. Imagine holding a piece of paper slightly bent but not folding or creasing it. The plant fibers that make up the paper will not have exceeded their elastic limit. So as soon as you let go of the paper sheet it will quickly return to its noncreased original flat state. However, if you were to roll that piece of paper into a cylinder and hold it for a few minutes, some of these fibers will be pushed beyond the elastic limit which is evident since it will not lie flat anymore since slight deformations have occurred in this sheet.
Now, when you properly fold a piece of paper as you would during Origami, the plant fibers along the crease are pushed into the plastic region of the paper, causing a fracture point at the actual line of the fold. A practical example of this is if you were to fold a piece of paper, you will note that if you stretch the paper evenly on both sides of the fold, the paper will tear right on the fold (a quick way to "cut" paper if you have no scissors). The fold then becomes an irreversible structural failure and the fibers in the paper will never regain their original state.
Because of this damage to its structure, the paper will from then on have this fold. And no matter how hard you try to flatten out the fold it will never return to its original state. This is why Origami models continually retain their shape.
A: 
Curved creases are sometimes used in origami – a practical example
being the French-fry box used in fast food restaurants. However,
little is understood about the mechanics of such structures. Now,
Marcelo Dias, Christian Santangelo and colleagues at the University of
Massachusetts, Amherst and Harvard University are the first to develop
a set of equations to describe the physics of curved-crease
structures. As well as providing a better understanding of origami,
the team hopes that the work will lead to practical 3D materials that
are both strong and flexible.
Santangelo and colleagues focused on a ring because it is a relatively
simple example of how a 2D structure can be transformed into 3D object
by creating a curved crease. To gain a basic understanding of the
physics, the team built a few origami saddles out of paper – from
which they deduced which physical properties are key to understanding
the mechanics of the curved crease.
At the heart of the transition from a 2D sheet to a 3D object are the
planar stresses created in the ring when it is folded. These stresses
are relieved by the sheet wrapping around itself to create a
saddle-like structure. If the ring is cut, then the stresses are
relieved and the saddle will collapse to a ring that will lie flat –
albeit with a smaller radius.

(Source)
