Quality of Light
The most important thing to think about when lighting a photograph is the quality of light you need. It is almost important to consider how best to achieve that quality. When thinking about quality of light, you first have to decide whether you want your light to be soft or hard. A light source that is small compared to the subject will be very hard, but a light source big when compared to a subject will give soft light. Hard light leaves hard shadows that sculpt and define a subject for an edgy look, but it can be harsh and unflattering. Sometimes hard light is exactly what you want, though, and a small light source further from your subject will do the trick.
When soft light is needed, such as for portraiture or product photography, a larger light is necessary. Remember we are talking about quality--not quantity--so the light does not have to be more powerful, just larger in comparison to the subject. Bouncing a smaller light off of a wall or ceiling is a great way to turn it into a larger light source. By varying the distance between your light and your subject, you can alter the quality of light falling on your subject and experiment until you get the look you are trying to achieve.
Direction of Light
The second most important thing to think about when lighting is direction. Where will the light come from in a photo is extremely important, as it dictates the pattern of highlights and shadows on your subject. It is important to get the flash a little bit away from the axis of the lens, as light coming from the side rather than straight on will reveal the depth and dimensions of your subject. This is why pictures taken with on-camera flash (such as license photos) often look so flat and dull. Many times you will find that you only need to move your light a little to the right or left of your camera to shape your subject, but experimenting with the position of your lights is key to coming up with new and interesting looks.
Using Additional Lights
Another thing to think about is using multiple lights on a subject. The main light used to illuminate a subject is called a key light, and often photographers will use additional lights in their setups. A second light is often used to put more light into the shadow side of a subject, and this is called a fill light. The fill light is usually less powerful than the key light and it is this ratio of brightness to darkness that defines the contrast of your photograph. After starting with a key light, experiment with adding fill and moving just the fill light around to see how you can affect your photo's shadow tones.