Here is an example. Suppose you have a dog running left to right across your photo. If his nose is right up against the right edge, he does not have anywhere to run so you put him in the opposite third and he now has all that space ahead of him to run into.
In the Nature Conservancy magazine is a photo of two turkey sized birds facing each other with their heads overlapping. It made a natural composition to leave them in the center, well-balanced, their heads reaching up and just in the upper half of the frame. It made nice lines leading upward forming a pyramid shape.Sometimes it is fine to have subjects centered.
Horizons usually work best when not in the center, a line across the middle of the frame. If the sky is dramatic, show mostly sky with the horizon line in the lower third. If it the foreground that is dynamic, have the horizon in the upper third.
However, if there is a dramatic reflection over quiet water, then it would be a nice composition to have the horizon line across the middle.
Someone wrote, if you go out of your way to break rules all the time that would be anarchy. If you never followed rules that would be chaos. So, most of the time follow the rules. Look at photos that please you and then analyze their composition.
In his book, "Photographically Speaking", David duChemin does just that. He critiques the composition of 21 of his photos.
In one of his photos he points out how it is not a very good composition, yet one that tells a moving story. He had to hurry to catch the moment, and it survived the broken rules.
Think of a stop sign near your home. The rule says you must come to a complete stop before moving on through. Hopefully, most of the time you do. Yet, other times you only come to a "rolling stop". And, I would be disappointed if you always got arrested for doing so. If you roared on through all the time, I would hope you would be arrested. See my point?
One more point about duChemin's book. He shows examples of the same image taken at different angles and views to help show you why he chose the final image as his choice. Photography magazines often do the same. Keep looking!