Aperture means "opening," and in photography the aperture is the size of the opening through which light is admitted to fall on the photographic medium.
Aperture is commonly referred to as "f-stop." The different apertures are referred to as (for example) f1.4, f2.8, f4, etc.
A wider aperture (f1.4, f2.8) naturally lets in more light, and the depth of field is much shallower. A smaller aperture (f22, f32) lets in less light and the depth of field is greater.
The question is often asked by beginning photographers, "How can I have something sharp in the front of my photo and the background be out of focus." This is done by setting a wide aperture and focusing on something close to the camera.
By contrast, a photographer who wants to have a large depth of field, with both objects close and far away in focus, will need to set the f-stop to a larger number.
Controlling your aperture gives you a great deal of control over your photos as well. You will be able to clean up backgrounds with a wide aperture, or include multiple elements in a photo with a small aperture.
When you make changes in your aperture, you will need to compensate by making changes in the shutter speed or ISO so that your photo will be exposed correctly. Shutter speed and ISO will be discussed in upcoming tutorials.
Curious about why smaller apertures are larger numbers and larger apertures are smaller numbers? Wikipedia has a great explanation here (involves math!).
Learn more about aperture at lovetoknow.com.
Check out the online depth of field calculator at DOFMaster.