After discovering James Gurney, and specifically this post, it's come to my attention that some of the stuff in the tutorial is misleading/wrong.
At its heart, composition is about information flow and digestion. What is going on, what you notice first, where you go next.
Good composition makes obvious which stuff is significant. It's about helping the viewer to understand what they're looking at. It paces the information so the viewer gets the main idea (e.g. setting, subject, mood) quite quickly, and then there's (usually) other stuff that they notice if they keep looking -- details, or mini-stories, or new interpretations, or whatever.
I say 'usually' because occasionally a simple composition can be effective even if it only has one main obvious idea. (In that case, I'd guess it causes the viewer to consider that one idea in more detail, perhaps using the picture as something for the eyes to rest on during contemplation. But now we're in Lulie's-wild-speculation territory.)
So when I said "you can control the way the eye moves", that's not strictly true. However, you can influence it. Specifically, you can make stuff more or less obvious/emphasised, using composition techniques (described in the tutorial) as hints to the viewer.
From Gurney's post (also see the other points there):
"1. The eye does not flow in smooth curves or circles, nor does it follow contours. It leaps from one point of interest to another. Curving lines or other devices may be "felt" in some way peripherally, but the eye doesn't move along them."
"Eye movement" itself doesn't necessarily work in the way I imply. Curving lines and so on are just indicators of what to look at. In practice the eye will be darting around, using these as clues about what's important in the picture.
It might be helpful to think of composition in terms of eye movement rather than information flow, because it can be easier to work out what will affect eye movement. (Both might be better, but eye movement is a good rule of thumb.)
Note also that this means everything is relevant to composition. If you have anatomy that's unusual, that will draw attention. If you have certain types of lighting, that will draw attention. If you have a meaningful interaction between two characters, that will draw more attention than characters with a boring or meaningless interaction. Composition can't be divorced from the subject matter -- every picture has composition, whether it's intentional or not.