If you look at well-known photographs (as opposed to photographs by well-known photographers -- that is, if you focus on the images themselves) you will find that most of them are well-composed. If you tear them apart, they have certain structural elements that make them visually engaging.
Some of these things are quite simple, and are virtual cliches -- don't have telephone poles, or anything else, growing out of peoples' heads, unless you're doing it for a good reason. If you have a portrait of somebody taken from the side, don't press his/her nose against the edge of the photo, while leaving space behind the head; that is, if there's space, have the person looking into it, not looking at the edge of the photo. Roads, edges and lines work best when leading the eye into the photo, rather than out. Simple things like these are almost instinctive, but they can be (and are) formulated as rules, and they can be learned, and they can be violated by people who know what they are doing. Other compositional propositions are not so much rules as ideas, but the ideas can be expressed and discussed and taken account of.
To say that photos have to be composed in a split second is only true in certain kinds of documentary photography (I would include wedding photography in that designation) and street photography, and in those areas, there is usually some leeway for poor composition, because of shooting conditions. And yet the very best street and wedding photos are well-composed. People who argue that you really don't have time for composition are correct in a sense -- you may not have time to contemplate a situation, or fuss with the composition. True. But if the composition isn't there, well, your photo won't be great, and won't become famous. Tough luck. The point is made clearest by the photos of Henri Cartier-Bresson -- he made possibly two of three dozen really famous photos (and maybe not that many), though he shot literally tens of thousands. All of his famous photos are well-composed; perhaps not because he composed them consciously, but because statistics were in his favor. That is, if somebody works hard at photography, and makes tens of thousands of photos, some are going to be well-composed, and those, if any, are the ones that will become famous. I think HCB became famous because he actually had such a high hit rate. Another person with a similar kind of compositional eye is James Nachtwey, who has actually been criticized because his best war photographs seem to be composed with an aesthetic sensibility that sometimes can make violence beautiful.
Sear Reid, who does the review forum, goes to a Daytona Beach motorcycle rally every year (if I recall correctly), sets up a camera and shoots passers-by. It's a form of street photography, but by choosing a background and an idea, he almost guarantees himself good composition. That is, he uses building elements and backgrounds to emphasize the movement and appearance of his street subjects. Many photo artists do that kind of things - and some of them create entire fictional scenarios that they then photograph, and these photos are usually beautifully composed. Most of these people know what they are doing; this is not happening by accident. They learned to compose photos, and not by random experimentation.