The rule of thirds is a well known fundamental photographic principle of good composition, and it's one of the first things that budding photographers learn in classes, and rightly so. However, it's only a guide, and ignoring the rule doesn't mean your photos will be dull or unbalanced.
Apparently studies have shown that when we view an image, our eyes do not immediately look at the centre of that image. instead we first look around the sides of the image. This means using the rule of thirds works with this more natural way of viewing an image. If you were to draw two imaginary lines at third intervals horizontally across an image, and then 2 more vertically, where they intersect is quite often the ideal position for placing your subject or an important part of the composition. Compare the above photo with the one below, and you can immediately see that the first one looks better.
Ok, I admit, that was cheating a bit because I chopped off the lovely looking sky there.
Bandstand in centre
Bandstand on a third
Even in a simple shot like this you can see that the first photo is more balanced (and looks almost interesting!). Here's another example:-
You can see the rule of thirds being used all of the time in movies and TV dramas, next time you watch a film or drama, make a mental note of where the main subject is placed, especially closeups of people, their eyes are usually on a third. Here's a screenshot of the BBC iPlayer, look at where the various celebrities are positioned!
Obviously someone at the BBC has visited this site and learnt about the rule of thirds!
And finally, keep the rule of thirds in mind as you edit your photos later on. Try cropping and reframing images so that they fit within the rules. Experiment with some of your old shots to see what impact it might have on your photos, remember it's only a guide, don't be afraid to break the rules.