Inside every camera lens is an opening called the aperture which works in exactly the same way as the pupil in a human eye. The aperture changes diameter from a wide aperture which lets in lots of light, to a narrow aperture which lets in less light.
Aperture diameter is measured by an "f-number". Typical values are f2.0, 2.8, 4.0, 5.6, 8.0, 11, 16, 22, 32. Somewhat confusingly, a smaller f-number represents a wider aperture and therefore more light, and a higher f-number is a narrower aperture and less light. (For the mathematically minded this is because f-number = focal length / aperture diameter).
A wider aperture/lower f-number (left) lets in more light, and vice versa.
An important thing to grasp is that these f-numbers are spaced one "f-stop" apart. By moving one f-stop you either double or halve the amount of light the aperture admits, meaning you need to either halve or double the shutter speed to keep a constant exposure.
Try to understand
The aperture is the size of the lens opening. It controls the amount of light let in: a larger aperture lets in more light, while a smaller aperture lets in less light.
An aperture is made out of aperture blades (usually five to nine) that form a rough circle to control the size of the opening, and therefore the amount of light let in. The size of the opening is that f/number. Basically, that f/number is a ratio of focal length to aperture diameter. Do you need to know this? Not really, except you should keep in mind that the aperture is a ratio.
Simply put, the smaller the f/number, the larger the opening (f/3.5 = 1/3.5 = 0.286; f/16 = 1/16 = 0.0625). That's all I'm going to explain here because it's an extremely complicated topic and you don't really need to know it. There's plenty of places on the web that explain it in great scientific detail; this is not one of them. The following is a diagram of f-stops. Each f-stop (or just a stop) lets in half the amount of light of the previous f-stop.
Now, lets get on to why this matters and how it can change what your picture looks like. Basically, a larger aperture (lower f/number) will have your subject in focus, and everything in front of and behind it blurry. A smaller aperture will have your subject in focus and everything in front of and behind it quite focused as well. Take a look at the following diagram; the further apart the lines are, the more out of focus something is:
Large aperture (f/1.8)
Small aperture (f/16)
Summary: Large aperture, background out of focus; small aperture, everything in focus. (Large aperture = smaller f-number, small aperture = larger f-number).