Aperture and f-stop are closely related terms. Aperture refers to the opening in the lens that light shines through when a photo is taken. A larger aperture obviously lets more light through. F-stop is simply the nomenclature that photographers use when discussing different sizes of aperture.
F-stops are usually given as "f/8" or "f/22." The numbers can range from less than one (only a few lenses and cameras are capable of f/0.95, for instance) to f/128. A higher f-stop indicates a smaller aperture and less light getting through. Usually, f-stops are indicated on a standard scale in which each increase represents an aperture that allows half as much light to get through. For example, f/8 allows half as much light through as f/5.6. While many cameras allow for f-stops that lie in between these standard f-stop settings, the standard scale looks like this:
f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32, f/45, f/64, f/90, f/128
Aperture is extremely important because photography is all about the manipulation of light. The proper f-stop for the lighting conditions is a major factor in the quality of the final photograph. It's hard to give specific rules for f-stop settings, because the right setting depends on a bunch of other factors, like the lens you're using, the shutter speed at which you're shooting, and the subject you're photographing. It will take some experimentation and experience with your particular camera setup to find the aperture settings that work best for you.
F-stops also allow photographers to manipulate depth of field to create different artistic effects in their photos. We'll discuss depth of field in detail later, but for now, note that a larger aperture (which has a smaller f-stop number) will give you a narrow depth of field, while smaller apertures (with larger f-stop numbers) will result in a large depth of field.