Now for the good stuff...
Composition
The Rule of Thirds
The Rule of Thirds is a very basic concept that divides the picture into thirds on both the horizontal axis and the verticle axis. The points where those lines intersect are what some call "sweet spots." If you place your subject on or near one of those spots, the viewer's eye go right to it, and then flow to the rest of the picture.

It's not a hard fast rule, but a very good guideline, and if you look at pictures in a newspaper or some pictures considered to be the best in the world, many of them have that basic element in the picture.
To center or not to center, that is the question
Most people will put the subject of their picture right smack in the middle of a frame. Personally, I avoid it like the plague. If you have a manual camera, it's easier to put your subject off center or at one of the sweet spots. If you're using a point and shoot, you really don't have that option because of the automatic focus mechanism in the camera. I find that placing the subject off center makes for a more interesting photo. There are certain times, however, when placing the subject in the center is THE way to go for a picture, particularly if you're shooting for a symettrical look.
Diagonals
When done well, placing things on a diagonal axis can make a wollup of a picture. Just run it from corner to corner and see what happens. Diagonals are fun to play with from time to time.
Using the frame wisely
What you see through your viewfinder is all the space you have available to make your picture. Some cameras have a slight allowance and have a smidgen more space around the edges than what is visible, but for most, what you see is what you get.
Allright, so why am I cutting off the guy's head?
Chances are, you're moving the camera when you press the shutter button. Try to hold the camera very still. Support the camera from underneath, not from the side. Also, make sure you're looking through the viewfinder when you press the shutter button. If it looks like your subject might be out of frame, step back, move the camera, but reset yourself before you take the picture.
Bending at the waist does not bring you that much closer to your subject to make a difference
If you are using a camera that has a set lens, particularly a point and shoot camera, once you think you're close enough for the picture, take ten steps forward. Lots of family photos tend to have a far-off quality to them. Most point and shoot cameras have a VERY wide lens on them. Nine times out of ten, the closer you get, the better. Bending at the waist is only going to bring you one foot at the most closer to your subject.
Now for some pictures
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