Despite often modeling real animals and objects, origami's goal is not to reproduce this world but to create another—one superficially similar, but that works by linear and planar rules. Keeping this in mind makes it easier to understand the way of folding (ore) paper (kami).
The place to start is Joseph Wu's Origami page; all this is found there.
For other general sites, Alex Barbur's has a searchable model database and the origami-l archives; the Origami Database and Diagram Database both claim to be completist; the former's forum is helpful. Origami USA and the British Origami Society are the main English-speaking organizations; both have print newsletters chock full of diagrams not found elsewhere, and the former a searchable model archive. The newsgroups rec.arts.origami and alt.arts.origami have a joint, limited FAQ. Gilad Aharoni reviews a basketful of books.
The most important specialized model site is, of course, these errata to the most popular books. There's approximately a metric elephant-load of paper airplane designs (and at least one simulator); I'm fond of Joseph Palmer's, though this Flapper is cool. Since geometry is intrinsic to folding, there's no end of math + origami sites; Origami Mathematics is a fave for its wicked intersecting tetrahedra—and there's always the fun of crease patterns. However, I'm more interested in polyhedra and other modular models. Dollar bill folding is expensive to practice, but it makes fun tips. Don't forget flower arranging, either. And when you start designing your own models, here's diagramming advice and a list of helpful softwares.
There are many artist sites aside from ones listed above, the most stunning being Robert Lang's. Of course, Lang's work is always stunning. Alex Bateman does some fascinating tessellations. Michael LaFosse's Origamido should also be mentioned. David Lister has several essays on the history and aesthetics of origami. And lastly, some people live origami.
And then there's supplies. Disclaimer: when Fascinating Folds folded up shop, I stocked up on gami goods, so haven't needed to try other mail-order sources (though I will soon). These look good, but I don't endorse a one of them. All have papers and books. Kim's Crane has a good rep; Shizu has loads tasty high-end goodies; Opane has an origoods section; and OrigamiUSA has a supply shop; look for others in Origami-L's list of suppliers.
And finally, in the not quite origami but still fun, there's other kinds of models: making your own envelopes, napkin folding, cut-and-folds (a.k.a. pepakura) from Paper Toys, Yamaha, or design your own (3D models not included), and, most startling, the working mechanical models of Flying Pig.