Rules of thirds

In the rule of thirds, photos are divided into thirds with two imaginary lines vertically and two lines horizontally making three columns, three rows and nine sections. The most important compositional elements and leading lines are placed on or near the imaginary lines and where the lines intersect.

By placing points of interest in the intersections or along the lines your photo become more balanced and will enable a viewer of the image to interact with it more naturally.

For example in a portrait the eyes may be lined up with the upper horizontal line and each eye is where the upper horizontal intersects with a vertical line. 

The main reason for rule of thirds is to show the points of interest and intentionally placing the lines in particular places. Also post production editing tools today have good tools for cropping and re-framing images so they fit within the rules.