Composition is the arrangement of elements within an image. It could further be said that composition is, beyond the raw selection of subject or subjects, one of the key tools that the photographer has in crafting an image.
Composition operates within the selected format limitations of the camera and recording medium, so an awareness of their basic properties are generally required. Format is generally expressed as a ratio based upon width by height in some measure, eg. 4x3 inches, 1920x1080 pixels, 36x24mm or 8688x5792 pixels. The important thing is the shape of the frame, not the units it is measured in. Many cameras allow you to alter this format within the camera, for example by changing the film type or altering a setting within a digital camera. It is always possible to crop (cut) a photograph down to a smaller format (permitting for some loss of quality), though it never possible to retroactively step outside of the bounds of the format at the time of capture. Composition is about fitting all the elements within that frame.
Lens and aperture
Many lenses are sharper in the center than at the edge, particularly when used under wide-open apertures. This is due to the optical properties of the lens, and may be a concern during composition. For example, if you have chosen to shoot at an aperture that results in a loss of clarity to the edge of your image, and the most important subject is placed there, then you will be very seriously influencing the overall aesthetic of the resulting image.
Modes of composition
In more considered modes of conception, such as studio photography, complete control may be given to the photographer to experiment with different composition options. In more spontaneous photography, such as event photography, the photographer may have very little time to achieve a composition for capturing a moment. In the former cases composition may be approached formally and in conjunction with other shot parameters such as camera position and perspective, aperture, shutter speed, lens, lens zoom, lighting, and recording medium sensitivity. In the latter cases, the photographer may already have an eye squinting through the viewfinder, and only a fraction of a second to re-frame the subject in order to capture some important moment.
The most widely criticized choices for composition are the following, though it is important to emphasize that they all have their place and can be effective and justified decisions at times, they are usually the most common failings within early amateur photography.
Moving the main subject off center takes a wee bit of forethought but it's well worth the effort. Most photography and videography follow the classic "golden mean" which divides the "frame", viewfinder or monitor, approximately into thirds vertically and horizontally (think tic-tac-toe grid). Main subjects and points of interest are best placed at or near the intersection of the lines. In scenics, place horizon either above or below but not through center.
Allowing too much in the view, such as legs and feet in portraits or group shots makes your subject too far away and small while including more distracting background. A famous quote states: If your pictures aren't good enough, you're not close enough! Think about composing from the hips up, or better waist up. Most people hold their camera horizontally and almost never think to turn it vertically. A general rule: take portraits of one person (or pet) vertically and two or more horizontally, along with the idea of "filling the frame" as professional photographers know it, will bring out intimate, powerful photos.
Using the portrait setting on point and shoot cameras or, on SLR's, using the larger apertures (the smaller numbers) allows the background to go out of focus and hold interest in your main subject. Also watch for poles, trees, etc. coming out of your subject's head or torso.
Lack of depth
The most common method is to include foreground such as close, aesthetically pleasing, rocks or flowers “anchoring” the photo. Tip: Eliminate flat subjects and "see" like your camera by closing or covering one eye. You'll notice how much easier it is to watch for clues and angles that suggest depth and counter flat lighting and busy subjects that meld into a confusing mess.