Origami models can, of course, be folded from virtually any paper—and indeed from other thin, flexible materials such as starched cloth, foil, and even metal. However, the processes involved in making many of the more complicated models (particularly the insects and other many-point forms) put strain on the material at the points and creases. With a little carelessness, it is very easy to tear the model while you’re folding it. (A few of the models on here have small rips hidden inside them, but if a rip runs along an exposed crease, there’s very little you can do other than throw it away and start again!)
Most paper I’ve found sold as “origami paper” tends to be 15cm square; it’s fine for less complex models but the detail required for the insects etc. just cannot be accomplished. The flaps become too small to manipulate in comparison to their length. However, I can recommend the 15cm paper found in Muji (see the post on the Hercules Beetle).
For the more complex models, larger paper is essential. I did find some large (34.5cm) paper on roze.co.uk (see Dragonfly) which is good, if slightly too easy to rip for my liking. (For some reason, the green used in the Angler Fish is less prone to this than the red or blue.) My latest discovery is 24cm paper from Origami-shop. I am very happy with this paper; it’s large enough to fold the complicated models, and strong enough not to rip. I also received some samples of other papers with this…
(To be expanded when I’ve tried some of the samples!)