The first thing is that there should be some point of interest in the picture that attracts attention. It may very well be an animal, flowers or, if you are doing a close up shot of an animal, its eyes. There could also be more that one point of interest and in many cases it is preferable to have more than one as it enables the eye to move around the picture. The eye can move across the photograph and, if you have elements that are close and far away, in and out of it as well. The placement of these points is very important and varying amounts of emphasis can be given to each point, depending where it is placed in the picture area.
If one divides the viewfinder of your camera up into 
vertical and horizontal thirds, the points where the two vertical lines and the two horizontal lines cross are very strong positions. These are ideal for the points of points of interest which you want to highlight while the lines themselves are useful guides for things like horizons. It is all too easy to just pop something into the centre of a frame, particularly on auto focus cameras.
The trick with cameras with only one focus point in the centre is to first focus the image and then, while holding the shutter release button down on the auto focus lock position, reposition the subject into the most pleasing position before pressing the button down fully and taking the picture. One may find that the placement of the subject in even more extreme positions, like in the outer columns, also works well.
In most cases when photographing subjects that have fronts and backs, like people, animals, cars and so on, it is often more pleasing to position them so that they have space to move into. Placement on the other extreme, so, that they are facing the edge of the frame. is rarely attractive but can be used to good effect when the photographer is attempting to create some tension in the image.
The placement of points of interest is of even greater importance when a wide angle lens is used. Wide-angle lenses tend to add the perception of space to a photograph and as a direct result the transition between close and far objects is greatly accentuated. This is why so many subjects seem to be small spots in the middle of the picture when point and shoot and other cameras with wide-angle lenses are used. The answer to this is to get closer (of course this is not always possible) or to put some other strong foreground element into the picture that will take the viewers eye to the subject.
When composing a picture in the viewfinder try to look at the image in the viewfinder as though you were looking into a slide viewer or perhaps at a photograph rather than the real live scene before you.
So often one is unhappy with the pictures after getting them back from the photographic lab and sees all sorts of unwanted things in the picture. The eye and brain are wonderful editors and while looking at a scene, will edit out all sorts of things, like telephone poles, bits of rubbish and so on. When looking at a picture on the other hand none of this editing takes place and all the blotches on the landscape are there in full view. Altering the way you look into the viewfinder will change this. Look at the picture in there and not at the scene in front of you. Look for things that seem out of place. Check the corners and sides of the frame. Are any important bits cut off! Have the points of interest been placed where you want them? Take it slowly. Look. Think.
