Placing the elements of your photograph off-center in the frame adds interest for the viewer.
Reason: Off-center elements give the viewer's mind-and-eye room to "travel" within the frame, and make the relationship between the subject and the surroundings more dynamic. (Whereas on-center subjects often appear more static.)
The Rule of Thirds is simply a Rule of Thumb for how to place your subjects off-center.
Specifically, the Rule encourages to to you mentally divide the frame into 3 layers from top-to-bottom, and 3 layers from left-to-right. And, having done so, to place the subject(s) within the layers, or on the grid lines.
For example, in a photograph with sky, place the sky (roughly) within the upper third, or upper two-thirds of the image, rather than at the mid-point.
Since its a Rule of Thumb, the Rule does not need to be followed slavishly. For example, several of the images in this collection use the Rule from left-to-right, but top-to-bottom is divided into halves rather than thirds.
See if you can tell how the Rule of Thirds has been used in-whole or in-part in this collection of photographs.
© Jerry Weimar Photography