The idea is that you can divide the image into 3 equal horizontal sections as well as 3 equal vertical sections, like so:
When the lines intersect that point is where you want your main subject to be. It could be the eye of the person you are photographing
or where the building is set
or where the bee is sitting on his flower
This is giving your subject more emphasis. You know the pictures that you or someone you know take with the persons head right smack dab in the middle of the picture with tons of head space above and little of their body? Not so appealing most of the time. I have, however, seen this done once where it totally worked and was beautiful. You have to know the rules in order to break them effectively. I do not always follow this rule of thirds, but most of the time I do and it is second nature. I did it even before I knew the rule. To me, it just looks better, and then I learned that it is a rule.
I mean it says "rule" but I sometimes like to break rules. If you CHOOSE to break this rule, just be careful and make sure it is still going to have the desired effect you want. I mean, play around with this and see what you like better. Ask others to critique your work for you, but when asking others be sure to ask someone with similar knowledge to get a educated critique. Sometimes I find when asking just anyone I get a lot of focus on how cute the model is instead of is this compositionally sound and does it draw you into the right spot, etc.
This is not the only rule in composition. There are others that help you to have a compositionally sound image.
Framing. You can use buildings, trees, balloons, or just about anything to frame your subject. Although, I think rule is too strong. This is a good tool and an excellent option for bringing your attention straight to your subject.
In this image I used the tree branches to frame my evil queen in my Snow White shoot. Your framing can be subtle or dramatic. I mean, you could have a frame that is an actual frame, or use a window. There are so many options. Try something new and maybe a little zany. See how it works for you.
Leading lines. In my very first photography class back in middle school we learned about leading lines. A leading line is something that creates a line leading your eye to your subject. An excellently composed image should be able to lead your eye directly to your subject. If there are more than one focuses in your image your lines should flow to lead you from one to the other and back again.
In this image below the box is creating the lines that draw your attention to the bucket which when you follow the circle brings your attention to the rings. Now for a ring shot you usually want the attention to immediately go to the rings but this one I wanted to take you all over so you can see the decorations as well.
Simplicity. Try not to over do it with information in your images. To much information is confusing. Your viewer will not know where to look and will likely just look away. So, when getting started in the photography world, keep in mind, the rule of thirds, leading lines and simplicity. There are a lot more rules and guidelines to photography but focus on these for your composition and you will not go wrong!
Here is another site with some information on composition. He adds more rules than me but I don't think color is a rule since I like black and white, but when shooting color it is a good idea to keep that in mind too. http://www.picturecorrect.com/tips/photography-the-rules-of-composition/