Any "Intro to Digital Photography" book will give you great introductory tips on composition for your photography.
I think a couple tips mentioned earlier are worth repeating. If you're taking photos of little kids, see if you can bend down to their level, and take photos from their perspective or from their eye level. Too often, parents take photos of their kids from adult height and adult perspective, pointing the camera *down* at their kids. Instead, try bending down to your kids' level, and capture their expressions from their eye-level.
The other big tip is something you've already touched on. Sometimes, the most interesting composition is when your subject is off-center, NOT when your subject is dead center. Photographers often refer to the
rule of thirds when giving introductory pointers on composition.
You can do a quick Google search on the "Rule of Thirds" for more information. Basically, you look in your viewfinder and you mentally divide the image into thirds vertically and into thirds horizontally. What you end up with is a tic-tac-toe pattern on your image. Here's a photo I stole from the Internet that illustrates this:
What do you do with this tic-tac-toe pattern? Notice that the tic-tac-toe pattern has
4 intersections (ie. where the lines cross). The Rule of Thirds states that the most interesting spots to place your subject is in any of the 4 intersecting points.
In this case, the photographer placed the subject (the woman's head) in the upper right intersection. Actually, her hand and the sun are in the upper left intersection. Together, this makes for a more compelling photograph.
Can you imagine if the photographer placed this subject right in the dead center of the photo? It probably wouldn't be as powerful or as compelling.
The other thing about the Rule of Thirds is that you can also place the horizon at either 1/3 from the top or 1/3 from the bottom to make the photo more interesting. Try not to place the horizon right smack in the middle of the photo.
In the above photo, you see that the photographer placed the horizon at the top third of the photo. This places more emphasis on the ocean and the beach. If you wanted the viewer to appreciate more of the sky and clouds, you'd place the horizon at the bottom third of the photo.
There are some other photo composition tips out there, like leading lines, etc. Here's an example of
leading lines (again, a photo I stole from the Internet):
Here, you see that all the lines in the photo (ie. the train tracks, the line of trees, the railing, the walls, the row of buildings) all lead the viewers' eyes right to the subject.
As you go out to take pictures, see if you can find other examples of "leading lines". Also, leading lines don't necessarily have to be straight, either. You can have curved lines that lead to your subject, like this example (again, not mine):
This photo also uses some Rule of Thirds technique, too.
Of course, none of these are hard-and-fast rules. These are easy introductory tips to make your photographs more interesting. But rules are sometimes meant to be broken. If your photo calls for your subject to be right in the dead center, then do what your photographer's eye tells you is right.
Again, check out any "Intro to Photography" book at your local bookstore, and they'll totally talk about the above photography composition tips.
Hope that helps!
