The rule of thirds comes from a theory that the eye is naturally drawn to intersection points that happen when an image is split into thirds.
Import focal points in the photo should sit at the imaginary intersection of these lines.
This gives you four focal points to place your subject into.
This focal point could also be referring to a horizontal third like a horizon or shoreline.
Let's take a look at an example.
My subject in this photo is the fishing boat. It is kind of lost in this scene. I'll fix this by doing some cropping.
Photoshop is kind enough to include the rule of thirds grid when cropping. This makes it a lot easier for me to place the horizon and my subject in one of the four interesections.
Not only do I want the boat to be the focal area of the photo, but I'd like to focus more on the water than the sky.
So the first thing I do is to make sure the boat is in one of the four intersections. Then I cropped the horizon to be along the top third of my photo.
The resulting image allows the eye to be drawn right into the fishing boat while also focusing on the water over the sky.
The overall composition of the photo has been vastly improved.