Let me start out saying that "The Rule of Thirds" is a compositional 'rule of thumb', not a hard and fast rule.
The first rule in photography is; that there are no rules.
To explain the rule of thirds, we need a visual. The rules says you should divide your image into a grid of thirds, like pictured below.
You'll notice the grid causes the lines to intersect at four (4) main intersections (Marked with a red circle). At the intersection of these lines is where you would place a point of interest for a subject. The rule of thirds is considered by many to make pictures more aesthetically pleasing and professional-looking. The rule of thirds can be applied by lining up subjects with the guiding lines, placing the horizon on the top or bottom line instead of the center, or allowing linear features in the photograph to flow from section to section.
The rule of thirds is a simple and effective way to produce aesthetically pleasing pictures, and once you start putting it into practice, you'll begin noticing it being used in paintings, billboards, and even television. The rule of thirds may turn out to be the most powerful tool in your camera bag.