The simplest things you can control on a camera are aperture, shutter speed and ISO (film) speed. Unless you are really trying to create a special look, you probably want to keep the ISO setting as low as possible to cut down on noise, so that leaves shutter speed and aperture. If you have a zoom lens (or two), you also get another lever to control: the focal length of the image. Here are two photo sets I built to help me visualize how the different settings will contribute to the final image. Use the prev and next controls to jump to the next setting, or press play to see a slideshow of the entire range of settings for the control.
Note: Thanks to Patrick Fitzgerald for creating the handy Slideshow Wizard, which is what I used to create these examples!
For this example, I kept the camera at a fixed focal length, focus and ISO setting, and used the aperture priority mode to automatically select the shutter speed. This shows how the aperture setting can be used to control the depth of field. The major trade-off is that the smaller aperture lets through less light, which means that the shutter speed has to be longer to achieve the same exposure. This is fine for shooting a static shot like this one, but would not work for moving targets. In those cases, you will need to increase the ISO speed (which makes the picture noisier), or add light (which is a whole different subject).
Varying Focal Depth
For this example, I varied the focal depth by varying the zoom lens (and switching lenses to cover the full range), and kept the subject size constant by moving my camera closer to the subject. The effect is to vary the separation between objects at different distances from each other. With a large focal length, the depth of everything appears compressed, and objects seem closer together. Conversely, with a short focal length, foreground objects appear much larger then background objects. This can be cool for making bobble headed cat pictures, or creating funny relations between subjects in a picture.
Equipment used: Nikon D40, Nikon 50mm f/1.8, Nikon 18-55mm f/3.5-5.6, Nikon 55-200mm f/4.5-5.6