The rule of thirds is a compositional rule. This rule is often compared with yet another guide is known as the Phi grid (also known as the golden ratio, sometimes also referred to as the Fibonacci Spiral).
The golden ratio is a bit too complicated to implement in composing but Lightroom has a way to incorporate that in your images. The rule of thirds, on the other hand, is a lot easier to understand. Cut the frame using two parallel sets of lines, one running from left to right and the other from top to bottom. The idea is to divide the whole frame into 9 equal rectangles.
There would be four intersecting points. The rule of thirds denotes your need to put the major points of your image on these four intersecting points to enhance the quality of your images. The eyes get drawn into these four points of intersection.