One of the things that first attracted me to origami as a child was that the tools were all around me; all I needed were my hands and a sheet of paper, and I could find paper anywhere! For many years, I used pads of obsolete business forms from my Dad's business, cut to square. Nowadays, the equivalent paper would be copy paper—used or unused. And all it needs is to be cut (or torn) to square, and you're ready to fold.
For simple folds, copy paper works very well. It takes a crease nicely and doesn't easily wrinkle. And you can find it almost everywhere. But as the world of origami has progressed over the last few decades, the complexity of origami designs has grown, and for many designs—and almost everything I compose these days—copy paper is just too thick. You'll need to look for other sources.
Even more ubiquitous than copy paper is magazine paper and newsprint. Both are tempting, particularly because they are generally much thinner than copy paper. But if you try them out, you'll find that magazine paper and newsprint are really lousy papers for origami. They have two big weaknesses: they are weak (and so tear easily), and even worse, they don't hold creases very well at all.
For many years, my paper of choice was a type called "second sheet" or "manifold." These names came from the days before copy machines, when business forms came in packets of several sheets with carbon paper between the layers, so that when you filled out the form you were making multiple copies. Manifold paper was quite thin (9 lb., using the American system of paper weights; see here for conversions), but unlike newsprint and magazine paper, it was crisp and took creases nicely. Alas, multicarbon forms seem to be a casualty of the commercial photocopier. Although I'm still working on the first 1000-sheet ream of manifold that I ever bought, it's been a long time since I've seen it in stationery stores. But airmail paper is very similar—thin, crisp, and strong—and email has yet to kill it off.
I regularly get asked, "do you have to use that special Japanese origami paper that comes in the little square packages?" It's true, there is paper made especially for origami, and most art or craft stores carry it. It's relatively thin, brightly colored (most packages contain an assortment of colors), and, conveniently, it is already cut to square. (Or nearly so; more on that in a minute.) It seems like this paper must be the ideal paper for origami, possibly made by hand according to ancient tradition and used for origami for perhaps hundreds of years!
The truth is somewhat less inspiring. What we think of as traditional origami paper—generally called kami—was actually developed in the early 20th century for use in schools, using inexpensive, western-style machine-made paper. It was colored on one side, because that's cheaper. The paper is not archival, and the dyes are unstable, so it degrades over a period of years. And while it is fairly thin (thus using less material, making it cheaper per sheet), it is not very strong, as generations of folders attempting to use it to fold insects, can attest!
And it is not always square, to boot. This is not necessarily due to imprecision in manufacturing, though. Kami, like all machine-made papers, has a definite grain, which comes from the manufacturing process. It is made on a continous belt moving through a paper pulp slurry, and the motion of the belt tends to align the paper fibers along a given direction. (You can easily see the effect of this alignment by trying to tear a sheet of newspaper; it will tear much more cleanly in one direction (with the grain) than the other (across the grain).) Paper absorbs moisture from the air, and a change in humidity will cause a slight change in the size of the paper. But because of the grain, machine-made papers swell more in one direction than the other. That means that even if the paper was square when it left the factory, it might not be square when you open the package if the humidity is different where you live.
So kami is not very stable, not very strong, and maybe not very square. Why use it at all? Tradition, and convenience. There is a history of using kami for origami; many folders honor that tradition by continuing to do so. And it's convenient to be able to just pull out a sheet and start folding, square or not. (And for many folds, being half a millimeter out of square is not going to make a significant difference.) Kami is most commonly available in two standard sizes: 6 inch (15 cm) and 10 inch (25 cm), and these are so common that if you ask any origami person, "hey, do ya have a square of six-inch on ya?" they'll know exactly what you mean. (At least in America, where we still use inches.) But if you look around, you can find it as small as 1" and as large as 15", and nowadays, there is an enormous variety of colors, patterns, gradations, and prints.
(A side note on terminology. The word kami is Japanese, and simply means "paper." The word "origami" is derived from oru, meaning "to fold," and kami, meaning paper. So, to a Japanese speaker, the word kami can refer to any type of paper. However, in English-speaking countries, kami is used informally in a narrower sense to refer to the pre-cut packaged squares, no matter where they come from.)
One drawback of kami is shared by most of the other papers I've talked about. It's very difficult to make curved surfaces. For the paper to take a crease cleanly, it should be a bit springy; and if it's a bit springy, and you try to curve it, the springiness will straighten out the curve. (There is a folding genre, called "curved tension folding," that overcomes this problem by creating locking mechanisms in the folding.) In the early days of modern folding, that didn't matter; most origami was flat. You could close every figure in a book, and so curves weren't even contemplated. But as people became more familiar with the works of the great 20th-century Japanese origami master Akira Yoshizawa, works that were delicate, 3-dimensional, organic, and curved, people realized that curved folds and 3-dimensional shaping were major elements of artistic folding. Traditional papers didn't allow this type of shaping (and Yoshizawa didn't use traditional papers for much of his folding). This led origami artists, individually and collectively, on a journey: the quest for the perfect paper! In the next couple of pages, I'll discuss some of the candidates.