Everyone knows the rule of 3rds, but I'm always shocked when I see compositions that would be vastly improved if the animator simply remembered this basic guide. This stuff can seem so obvious, that we forget to think about it! most of us just assume our brains are wired to know these type of things intuitively.. but we ALL need to be reminded sometimes, for the benefit of our artwork. It also helps to know a little about WHY it works..Simply put, if you divide the field into 9 sections by drawing two vertical and two horizontal lines, you should put things of interest on the lines or around the points where the lines intersect. I think you will find that some of your favorite compositions follow this guide. (I like the word "guide" instead of "Rule" or "Law") And of course, just to head off the hordes of defensive over-individualists out there, rules are meant to be broken, and often should be within certain circumstances.. but really... i bet your shot will look better if you just stop talking for a second. Art is a form of communication, and you're not going to get anywhere speaking jibberish.
The rule of 3rds seems to be derived from Leonardo's "Divine Proportion", the visual representation of a + b over a = a over b = Phi (1.61803...). In simpler terms, imagine a line divided in two pieces. This formula states that the whole of the line is to the larger section as the larger section is to the smaller section. The "Divine Proportion" is all around us, from art to nature to music to our bodies, to just about anything.. nautilus shells, the cochlea in the inner ear, the pyramids of Giza, snowflakes, spiral galaxies, even the music of Beethoven and Mozart!Back to using this idea in film and art... above, "Touch of Evil" used it to perfection throughout. Within illustration, it would be hard to find a single Norman Rockwell that isn't composed effectively using the rule of 3rds.
Of course you should break this guide from time to time, especially if you do it to further the characters, or the shot by shot direction, for example if you centralize a composition in order to make the main character feel like a target! there's always good reasons to break it! I like to turn the guide lines on in After Effects, and slide my artwork around until it hits that sweet spot! I do the same in iphoto, where I very rarely don't crop or move around a snapshot.