Sorry that the phrase "Tool of Thirds" falleth not trippingly from the tongue, but the point was clarified long ago by the famous American photographer, Ansel Adams, who wrote:
"There are no rules for good photographs, there are only good photographs."
It may be possible to tease apart the rules in the following image, but it's complicated, may be more valid in this case retrospectively than prospectively, and really: who cares? I simply think that it looks good.
So, if there are no rules, then what's with all of those rules? Allow me to list a few, often stated dogmatically to neophyte photographers:
the rule of thirds;
avoid telephone poles and horizons through people's heads;
fill the frame;
diagonals add energy, horizontals and verticals add peace;
look for S-curves;
avoid large areas of black;
bright areas up, dark low;
and many, many others.
The first "rule", too infrequently stated, should be to think about light. You can compose masterfully, but as a friend replies, compassionately and lovingly, when his wife points out something to stop and photograph: "Wrong light."
"Light" is a topic of its own - more some other time, perhaps.
Also, check out a brief video on "tips" for taking better photos.
The Rule of Thirds means to keep the subject out of the centre. This may be harder to do than it used to be before autofocus, which sometimes focusses near the centre, so it's too easy then just to shoot. More often, you will want to control the focus, and then compose it the way you want it ... often off-centre.
Vaguely related is the prohibition against bisecting the photograph, most commonly done with horizons; generally, framed images look better if not split in equal halves.
Telephone poles: what this really speaks to is that photographs are two-dimensional, so the pole looks like it's growing out of someone's head - same with lamp-stands, trees and, in the other direction, horizons.
The more general principle is that things which the human brain understands and filters - out or in - are not treated so mercifully in a photograph, and one of those things is the third dimension: depth. The third dimension can be simulated, but obviously it is not literally present in a conventional photograph; you can't stick your hand into it and reach further back. The image is flat. The depth in the following photograph, for example, is a completely illusory effect.
Fill the frame: sometimes, the context of a photograph matters, and is what you want to convey. More commonly, we inadvertently include peripheral and tangential elements which do not contribute to, and actively detract from, the image. If in doubt, close in. Figure out what you want to photograph - what you want to "say", what "story" you want to tell - and get rid of the rest.
The soon-to-be-famous Low Rule is a special case of frame-filling: Avoid Scruff. Brown twigs and grasses surrounding your waterfall are horrible, do not add context, and should not be there. The version below of the photo above is as I took it - quite nice, but I think that most of us would agree that it's just a bit better, in the first one, by closing in a bit (on three sides, if you look closely), leaving enough for context - but barely.
But this iceberg shot contains lots of "environment", which is part of what the photographer (me) wanted to convey. And ... it uses white space, not black space - see below.
Diagonals are fairly self-explanatory: as a "rule", they look good. If you wish to convey tranquility, keep things more vertical or horizontal. Or, combine them:
That is more pleasing because of the slight diagonal of the fencepost (and pitchfork tines).
S-curves just look good! Our brains like them. We more iconically see a road or path, winding through a meadow or a forest, but the principle applies to anything. Once in a while, we just have to undulate!
Black: large, featureless areas in a photograph can be very effective, referring more often however to "white-space" than to "black-space". It will happen but rarely that you truly want such a dark, lugubrious-feeling photograph as provided by large areas of black.
(To the goth-photography stylists: sorry; I just don't get it.)
And often, the black is inadvertent, because shadows which had detail to the eye will often go black to the camera. Cameras exaggerate contrast. Think "black eye-sockets" on a bright, sunny beach; we've all seen those pictures.
There are techniques to "bring out the shadows", both in the camera and in post-production, and when you need them, you need them. The better the photograph you make in the camera, the more effectively you can edit it. So, use techniques for avoiding such extreme contrast at the moment of exposure, e.g. photographing at dawn, waiting for a cloud, using fill-flash or reflectors, and many others.
Rarely, a lot of very dark acreage works.
More typically, think about how to bring out a little detail in the shadows.
Bright up, dark down - often, it will look better to darken the bottom of your photograph and lighten the top.
The pavement was darkened in post-production - it helps to propel it from a "nice" photograph to a "wow" photograph.
My bad: I've just done what I decry, by prescribing rules. None of these are "rules"; they are, however, very useful "tools". Breaking these rules is completely allowed, encouraged even, but do so knowingly, skillfully, intentionally ... have reasons, and they had better be good reasons. Whether you're taking minutes to set up a still life or a landscape, or seconds to grab a passing moment for a snapshot, knowing the rules will help you.
The eyes have it: there is something very primal, hard-wired into the human mind, about eyes. Apparently, a baby will look at three points making a triangle ... only if it's pointy-end down. It's looking for eyes and a mouth, presumably. We retain that into adulthood.
Your camera may have some variation of an "eye-focus-priority" feature. Use it!
It almost never occurs, in portraiture, that you want the eyes out of focus. Circumstances may dictate that you can't get both the eyes and the nose in focus, so choose the eyes. If the two eyes are different distances from the lens, choose the nearer eye.
Eye-sharpness may be the exception to the rule ... about no rules. If anything in photographic composition is truly even close to a rule, it's to get the eyes in tack-sharp focus.
Apart from that, they're all "tools" - not "rules"!