Ori (fold) gami (paper) reportedly dates back to the fourteenth century in Japan, when it was supposedly developed as an activity which would help girls prepare their fingers for sewing (1).
Many people think automatically of cranes when they think of origami. Cranes are a symbol of longevity in Japan, and there is folkloric belief that folding 1000 origami cranes will help a sick person get better. This is known as senbazuru, which comes from sen (1000) ba (a counter for birds) and zuru (crane). After World War 2 schoolchildren from all over Japan brought senbazuru strings to the atomic bomb site in Hiroshima. This practice continues today, and provides a hopeful spot of color at the otherwise grey and sombre site.
Almost any animal or object you can imagine can apparently be made from origami, as this beautiful swan suggests. Some people specialize in folding dollar bills or 1000 yen notes into origami creations, or fashion temporary hashioki from the wrappers for disposable chopsticks.
(1) Dalby, Liza, et al. All Japan: The Catalogue of Everything Japanese. New York: Quatro Marketing Ltd., 1984, p. 184.