Origami is the art of folding paper. The name comes from Japanese, adopted into English in the 1960s.
Before 1900, there were several isolated folding traditions which seem to have developed entirely separately:
In Europe, the first folding traditions were not with paper but with cloth. Napkin folding traditions appeared in the 16th century, and were very well documented, though aside from some simpler forms they were discontinued during the 1800s, due to the time and effort involved in folding cloth. Paper folding emerged as a form of children's recreation and a pedagogical tool, particularly during the 19th century and later. European folding generally uses 45º angles, and used mostly squares or rectangles. 
Chinese folding traditions are for the most part associated with the practice of burning paper objects at a funeral, as a way of providing for the deceased in the afterworld. As a result, there is an emphasis on inanimate objects; gold ingots, or yuenbao, are a common subject. Other uses of folding included auspicious animals such as the frog or turtle.
Traditional Japanese origami had its beginnings in the formal gift-wrapping practices of 17th century Japan, and by the mid-18th century it was an established and recognizable tradition. Several origami books were published in Japan during the early 19th century. Contrary to modern practice, it often used cuts, and implemented a variety of starting shapes: squares, rectangles, octagons, hexagons. Japanese traditional repertoire uses 22.5º angles extensively, and used more sculptural shaping than other traditions.
The pedagogical methods of Friedrich Froebel, developed in the 1840s and 50s, included paper folding to teach geometry and encourage imaginative play through interpretation of folded objects. During Japan's industrialization process, parts of the German education system, including Froebel's kindergarten method, were adapted for use in Japanese schools. The folding styles were assimilated into Japanese origami, and laid the ground for future developments. The restrictive use of squares with color on one side is credited to Froebel's influence. 
During the early 1900s, several Japanese folders started creating and publishing their own origami designs, notably Akira Yoshizawa, Kosho Uchiyama, and Toshie Takahama. Akira Yoshizawa's innovations in particular sparked an era of expanded exploration, and many origami artists credit him as the founder of modern origami. He was the first to create origami as sculptural art, and introduced a number of major technical innovations: backcoating, where two sheets are pasted together to combine the properties of both; wetfolding, where a sheet of thick paper is dampened before folding to loosen the fibers, allowing for better manipulation of the paper; a method of diagramming, which was expanded upon by Samuel Randlett, and now referred to as the Yoshizawa-Randlett diagramming system; partial folds and creases, used for sculptural effect; and in general a greater level of complexity and liveliness.
(more to come: origami after world war II in America and Europe, technical origami movement, modular and tessellation genres, etc.)
- Paper Fundamentals
- Joan Sallas, "Gefaltete Schönheit"