The aperture setting is a setting, like the shutter speed setting, to control the amount of light which reaches the film (film cameras) or CCD chip (Digital cameras). The larger the opening the more light is allowed through. The smaller the setting the less light is allowed through. Below is a chart which illustrates possible aperture settings of a camera
Okay, now things get a little confusing. Shutter speed settings make sense since we all know that 2 seconds is longer than 1/500th of a second. However, aperture settings often confuse beginner photographers because the numbers seem counterintuitive to logical math thinking. You have to remember with aperture settings the larger the number, the smaller the hole and the smaller the number the larger the hole. Say this over and over to yourself as many times as you need in order to remember it. This is a very important manual photography setting and understanding it will allow for much great control over the outcome of your final product.
Your aperture setting also controls depth of field, which simply put is the distance that will be sharp, or in focus, from your foreground to your background. The larger the opening of your lens the less depth of field you'll have the smaller the opening the more depth of field you'll have. The diameter of the aperture is measured in f-stops. Depending on the camera you're using or the resources you're reading f-stops will be displayed a little different.
f1, f1.4, f2, f2.8, f4, f5.6, f8, f11, f16, f22, f32, f45, f64, f90, f128
Here is a video explaining f-stops:
Here is a video explaining Aperture
(video by B&H)
Like the shutter speed, changing your aperture settings by one f-stop will effectively double the amount of light coming through (or cut it in half depending if your opening the hole more or closing it). An aperture of 2.8 will let in twice as much light as an aperture setting of 4. At the same time an aperture setting of f5.6 will let in half as much light as an aperture setting of f4.
Let's take a step back now and look at a couple of photographs which will show you how aperture settings affect the depth of field of a picture. In the following example, the photographer used a shallow depth of field by using a larger aperture opening (small f-stop number). The result is a main subject who's in focus but a blurred background.
Here is a video on this subject: