The aperture is a small opening behind the lens. It is made up of a number of leaves that slide over each other so that the opening can change size - you might recognise its design from the start of the James Bond films.
In an ideal world the aperture would be a circular hole, but often it's a regular polygon. You can sometimes see the shape of the aperture in out-of-focus lights, for example. The appearance of out-of-focus highlights is called the bokeh, and some lens are complimented on this quality in reviews. The aperture is inside the lens on an SLR camera, so the shape - as well as the range and number of settings - might vary from lens to lens in your collection.
The main reason you would change the aperture setting on your camera is to adjust the depth of field. To get the correct exposure, though, you will need to select a matching shutter speed.
Another reason for changing the aperture is that the performance of a lens might vary according to the aperture used. Almost all lens perform less well (this means that they won't be so sharp) at the extremes of the aperture range, so - if quality is important - you might want to use an aperture setting in the middle.
Of course, because correct exposure requires matching the aperture to the shutter speed, if you want to use a specific shutter speed, then you will need to choose an aperture to make that shutter speed possible given the light conditions.
The aperture is usually measured in f numbers . Confusingly, the smaller the f number, the larger the aperture (i.e. the hole) - so f4, for example, is a larger aperture than f16.
The other thing to be aware of is that to decrease the aperture by one stop (see exposure), you need to multiply the f number by approximately 1.4. For example, one stop down from f8 is f11, not f16 - or, if the correct exposure in a given situation is a shutter speed of 1/200s and an aperture of f8, the correct aperture for a shutter speed of 1/100s would be f11.
The reason for this is that the f number corresponds to the diameter of the aperture, whereas the amount of light getting through will be proportional to the area of the aperture. As area is proportional to width squared, to double the area you need to increase the diameter by the square root of two, which is approximately 1.4.