I can summarize my “rules of composition” thusly:

Placing your subject in the center drains the energy from the shot
Try to pull the viewer’s eye into the shot, not out of it
Look for interesting patterns
Try to use light and line to enhance your motif
Break whatever rule you need to

I think I’ve been blessed with a way of looking at things that produces images that other people find interesting (except for those pictures of the ground).  So a lot of what I do is unconscious.  But I’ve read some books and the above points seemed quite reasonable to me, so I think they come close to “codifying” what I do.  These points apply most strongly to non-portrait shots.  I’ve yet to come up with a rule set for portrait work.

For most shots you’ll have a subject. If you don’t have a subject you might re-evaluate whether you have a shot.  Your subject might be a mountain, flower, grove of trees, dramatic sky, river, vase, dog, pattern on a rock, or something else entirely.  At times you may find yourself wanting a shot of a particular scene because it’s beautiful but there’s no discernible subject – at such times you might realize that, while standing in the scene you experience its beauty but as soon as you put a frame around any part of it you can’t get the shot you want.  At such times I usually just sit back and enjoy the moment.

Placing your subject off-center in your shot introduces asymmetry and “tension”.  Placing the subject in the center of your shot does not.  Think about what you want.

The ever popular “rule of thirds” says that you should draw a tic-tac-toe board over your shot and place your horizon line on one of the horizontal divisions and your subject on one of the intersections.  This isn’t a bad place to start.

Your shot is composed of lines and light.  When thinking about the lines in your shot it’s generally a good idea that they lead “into” the shot, not “out of” the shot.  Otherwise the shot doesn’t hold together well and looks “disjointed”.

Sometimes what catches your eye isn’t any particular subject but rather a pattern of lines and light.  Sometimes this works, sometimes it just goes away when you try to frame it.  I haven’t discovered any rule of thumb, so I just keep my eyes open for it and try.

Every shot has a “motif”.  My friend Chris was really in tune with this and it kinda rubbed off on me.  I am not 100% sure what he meant by it, but what I took it to mean was a story, or a general theme.  Try to be sensitive to what your motif is, and then select the other aspects of your shot (where your subject is, what patterns you use, where your lines go, what sort of exposure you use, etc) to enhance this motif.

Finally – as with any other creative art, it’s important to know the rules and to be able to evaluate an idea with an eye toward either following or breaking the rules.

The wonderful thing about digital photography is that it doesn’t cost you anything to experiment.  You can indulge your whim as often as you like – just try to learn from your experience so you’re always improving.