From Wikipedia:
The rule of thirds is a compositional rule of thumb in visual arts such as painting, photography and design. The rule states that an image should be imagined as divided into nine equal parts by two equally-spaced horizontal lines and two equally-spaced vertical lines, and that important compositional elements should be placed along these lines or their intersections. Proponents of the technique claim that aligning a subject with these points creates more tension, energy and interest in the composition than simply centering the subject would.
Rules are like standards; the nice thing is, there are so many of them. Sometimes I use a Rule of Halves, or a Rule of Fourths, although I try not to be ruled by fifths. (Old rockabilly lyric:
"I've got four on the floor / and a fifth under the seat".) Sometimes I subvert the Rule of Thirds by filling the central ninth and leaving the rest sparse, or vice versa.
Ah, so many rules and techniques. To see which apply and how, look at some comic book and other magazine covers; at billboards, posters, display ads; at dioramas, tableaux, clusters of statuary; at architecture, classical and modern. Read books on display typography and page layout. Hint: many rectangular pages more closely follow a rule of 3/4's or 3/5's, or 2/3's or 2/4's, as columns/rows. I seem to see the Rule of Thirds mostly in square formats, and not even all of those. Look at covers of square media products.
The purpose of any graphic presentation is to grab eyeballs. Compositional rules are supposed to drag your attention towards whatever the artist wants you to notice. If clever, the artist may include contrasts and contradictions and ironies at other points in a compositional structure. The invention of perspective in the 1300's directly addressed the problem of directing eyeballs to desired locations. These principles were often overlaid upon earlier compositional rules like the
Golden Mean (don't worry, I won't quote it here) of which the Rule of Thirds may be a mere bastardization. Theories of leading lines, following lines, all that stuff, IIRC.
Whatever works for you, do it. Sometimes having all the action occurring along one edge of a wide image, or two edges of a square image, generates considerable impact. I recall one album cover (remember 33 rpm albums?) that was mostly blank, with a low city skyline along the bottom and half a looming skyscraper along the right edge. Titling was a small block in the far upper left. More like a Rule of Eighths there. Putting the subject at a far edge of a 2:1 pano implies the dynamism of entry or exit. There are many ways to grab attention. A bullet hole in the exact center works too.
Look at some modern works on visual design. I don't think the Rule of Thirds has a prominent part. But I could be wrong. That happens sometimes. Like day before yesterday.
Oh yes, the photos above. Ummm, where is the subject? Where is my attention supposed to be directed? The picture needs a stegosaurus or flying saucer or something. OK, it's shooping time!