Origami itself is known as the Japanese art of Paper Folding. It actually was 'invented' in China around the first or second century, and migrated to Japan around the sixth.
It is believed to have spread to Europe through the Moors, to have appeared in Spain somewhere between the 10th and the 13th century, and to have spread further in Europe as time progressed.
It finally reached the United States and the United Kingdom in the 19th century, where it gave birth to Mathematical Origami.
A better history is available at http://tqjunior.thinkquest.org/5402/history.html?tqskip=1
In 1945, in the United States, people started to look at origami from a mathematical point of view. From there, the great adventure started. People started to publish axiom sets, proofs about origami models, and other such things.
The most powerful set of axioms known today is from Humiaki Huzita, and was published in his paper "Understanding Geometry through Origami Axioms", published in "Proceedings of the First International Conference on Origami in Education and Therapy". The conference was held in Italy, in 1992.
A better and more thorough history is available at http://www.paperfolding.com/history/
Computational Origami is Origami seen by a computer scientist. That is, Origami explored with the purpose of designing algorithms to solve Origami problems, proving the universality of some problems (proving that a solution always exist) and the intractability (cannot be solved by computers in a reasonable amount of time, for non-trivial instances of the problem) of other problems (or worse, of the same problems!)
The current work in Computational Origami is well summarized by Erik Demaine, at his webpage.