The Rule Of Thirds is a technique many photographers use to make their photos more engaging to their audience. The theory is that if you place points of interest in the 4 intersections or along the lines that your photo becomes more balanced and will enable a viewer of the image to interact with it more naturally and become more engaged.
SOME EXAMPLES:
As you can see from the examples above, the points of interest or main parts of the image are on an intersection or across a line.
Examples of my work:
The photos above have been cropped so they have a more aesthetically pleasing composition.