Far from just being a children's craft, origami has been around for centuries and is practiced in cultures all over the world. Just about every craft shop has some kind of basic origami kit for sale, although there are more specialist retailers that can provide origami supplies for those wanting to experiment with different paper weights, textures and shapes.
For beginners, any square sheet of paper is good enough to practice basic folds and scoring techniques. Proper origami paper though (or "kami" to give it its correct Japanese name) is specifically made for the purpose of origami. As well as being able to choose your preferred colour or pattern, a good origami supplier will stock a large range of different paper sizes and weights which should be chosen based on what kind of sculpture you're looking to produce.
If you're looking to create some real origami masterpieces, sooner or later you'll likely to need some additional origami supplies including clips, tweezers and bone folders. Rulers and embossers also come in handy for creating super-sharp creases - and if your budget allows, keep an eye out for foiled-backed or gold leaf paper that will literally make your creation sparkle.