You can read Brad Warner's description of the formal process of doing zazen (坐禅 Chinese: "zuò chán") at the Dogen Sangha page, also linked by at his blog -- which I recommend by the way, both the instructions and the blog in general -- but here's an explanation of the sort of simpler form I first learned:
First, you find a place to sit. At least at the start, it's best to find a quiet, out of the way place where you won't be interrupted. You sit comfortably in a way you can keep your spine straight -- which to most Western people used to sitting in chairs will feel like you're arching your back a bit. Traditionally, we do this sitting on the floor with a little bit of a cushion under the tailbone. In Japan, this is often a zafu, or round cushion, and a zabuton, which is a flatter square cushion. (As usual, it sounds much more interesting and exotic in a foreign language: it really means "soft cushion" and "sitting cushion" and it's basically the traditional Japanese sitting on the floor version of "chair".) For myself, I've found that a yoga bolster is a little better support; the Boulder Buddhists use a square block cushion that some people find convenient.
Buddha, of course, used a patch of dirt, and I've discovered that if you find a small rise so your legs can go down a little, it's actually a very comdforable way to sit on the ground. Ideally, you should sit in padmasana, "lotus position" in yoga. I don't know about you, but I haven't been able to sit in full lotus since I was eleven; I sit with my legs comfortably crossed.
Sort of rock around a little bit and you'll find yourself settling down; push your chest out and your shoulders back a little to get your back really straight. Again, there are lots of formalities involved, but the point is just to get your back comfortably straight; when you find the spot, you'll feel very solidly attached to the ground, with little urge to rock or move. I can actually sleep in that position without a back rest.
Of course I've been doing this for getting close to 50 years. If you're just starting, and you're used to sitting in chairs, you may have a little trouble with it at first. That's all right; you can use a chair if you like. (These instructions are good for other positions.) The point of the position you sit is really simple: it should be a secure, comfortable way to sit that lets you breathe freely. It's important not to slouch, because that makes it harder to breathe; it's important to feel comfortable and solid and secure because if you don't, it's too easy to find yourself dwelling on thoughts about falling over. But the point of these positions isn't to prove you are a good ascetic who can bend his body to his will -- it's to settle your body down safely so you can pay attention to other things.
I am, by the way, a little bit of a heretic in this: other people find the ritual aspects important, or find the practice of overcoming pain to be helpful. Me, I think you can expect some discomfort, especially starting out, but actual pain is a hint that what you're doing that hurts might best be avoided.