The Rule of Thirds is applied by aligning a subject with the guidelines and intersection points of a grid. It means that the subject isn't centred in the image, which is how most people take their photos. Using the Rule of Thirds means that the viewer's eyes are drawn into the composition instead of just glancing at the centre. 

You can try this out by drawing the above grid over some of your photos. If they do not fit the Rule of Thirds, then you can simply crop it so that it does. This is a simple way to make your photos more eye-catching. 

The photograph of the cheetah above is following the Rule of Thirds while the photo below doesn't. Which do you prefer? Personally, I think the image above has more interest because it makes me look at the whole picture and composition rather than just the subject.