A simple way to improve the composition of your photos is to follow The Rule of Thirds. Divide your view into thirds, both horizontally and vertically. Imagine drawing horizontal and vertical lines so that you have 9 evenly sized rectangles. The 4 intersection points are your optimal points for placing your subject. More and more point and shoot cameras now offer a feature where you can turn on gridlines on your display screen. This helps to quickly compose your shots.
The Rule of Thirds helps you to compose well balanced shots that will make your photos more interesting and more pleasing to the eye. This same technique is often used by artists. Even though it's called the Rule of Thirds, keep in mind that it is more of a "guideline" than a fixed "rule". Depending on your photo, sometimes the Rule of Thirds should be ignored. Each photo is unique and should be composed in a way that suits the photo.