Digital pictures are the result of light hitting the camera's image sensor. Too much light and the picture will be washed out. Too little light and your photo will be too dark. The shutter controls how much light enters the camera. There are two settings which are related to the shutter: aperture and shutter speed.
Your cameras aperture is the size of the opening between the lens and the image sensor. Large camera apertures allow more light to enter the camera than small apertures. Apertures are expressed in 'f' numbers -- the higher the number the smaller the aperture. Standard lenses are usually rated between f/1.8 and f/16.
Each f-number allows twice as much light to enter the camera as the previous f-number. For example, f/8 produces an exposure which is twice as bright as f/16.
Camera aperture settings have two basic effects -- the amount of light which strikes the image sensor, and the 'depth of field'. Depth of field refers to the length of the image which is in focus. Large aperture settings have a shallow depth of field -- this means that the focus of an image is relatively short which causes foreground and background objects to appear out of focus. Small apertures have a deep depth of field -- almost all the objects (foreground and background) will remain in focus.
The aperture is directly related to shutter speed for determining the amount of light that enters a camera. Large apertures combined with fast shutter speeds let in the same amount of light as small apertures combined with slow shutter speeds. Determining which combination is best for a particular situation requires pictureic judgment that comes with experience.
To make it easier, most cameras have an automatic setting which will do the calculations for you. Many pictureers, however, wish to control aperture and shutter speed for artistic effect.
Since a large aperture can be used in conjunction with a fast shutter speed, this is often a good combination for action shots because the fast shutter speed will 'freeze' the motion with a minimum of blurring. Large apertures can also be used for low light conditions where there is very little movement in the scene. In this situation you would combine the large aperture with a slow shutter speed.
Simple point-and-shoot cameras usually have a fixed aperture, and it is only with more expensive models that you have adjustable aperture settings. When choosing a digital camera one consideration should be the range of the aperture. There are several ways this can be expressed: maximum aperture size, range of the aperture, maximum wide-angle and maximum telephoto apertures.
It is probably more useful to know a particular cameras aperture range rather than the maximum aperture size. A larger range gives you more flexibility in the kinds of shots you can take. A good range for all-purpose picturey is from f/1.8 to f/16.
Each camera lens has its own aperture rating. Telephoto lenses typically have a shallower aperture range than wide-angle lenses because longer lenses need proportionally more light. This is because they are gathering light from a smaller source, so larger apertures are needed to produce f-numbers which are consistent with shorter lenses.
