The rule states that:
1. an image should be imagined as divided into nine equal parts by two equally-spaced horizontal lines and two equally-spaced vertical lines as shown below:
2. important compositional elements should be placed along these lines or their intersections. See examples below:
The focus in this picture are the birds. See how the two birds are placed at the intersections of two lines.
The focus is still a bird. See how the body is placed along the line to the right.
Placing the element of interest on one of the intersections or along the lines will create a balance and interesting look on viewers' eyes. If we put the object of interest in the center, it tends to create a monotonous and static feels. Same thing happens when you place a horizontal element in the middle of a picture.
One thing to remember is, points of interest in the photo don't have to actually touch one of these lines to take advantage of the rule of thirds. Falling near the intersection of two of the lines, close enough to take advantage of the rule.
But you know what people say about having a rule. It's meant to be broken. Sometimes, to achieve a good composition, you have to break the Rule of Thirds. So, don't be too stringent on yourselves. It's okay to break this rule once in a while. It's just here as a guide not to bind you for the rest of your lives.