After writing this, I'm not sure it came out as simply as I'd hoped. Never mind...
Think of the aperture as a circle and the f number as a fraction. You are changing the size of that circle. I find it easier not to read it as f/2 or f/4 - instead read "one half (of the theoretical maximum diameter)" or "one quarter" (or one eleventh, etc.)
Each "stop" of light is a halving or doubling of the *area* within the circle, which is why the numbers for each stop aren't linear - the relationship between the diameter and area of a circle is not linear. This is how you control the exposure (along with shutter speed and ISO).
A "side effect" of changing the size of this circle is that it is one of the parameters that defines your "depth of field". The other factor you have any real control over is the focal length. Time for another explanation!
If you ran a tape measure from the camera off into the distance, you can focus the lens at pretty much any point on that tape. The depth of field is the distance in front and behind that point which is also "acceptably" in focus. The smaller the aperture circle (smaller *fraction* - 1/11 is smaller than 1/4), the larger this area is.
Why do you care? Well, imagine you have a group of people at differrent distances from the camera, you might want a large depth of field (DoF). Similarly, you might have someone you want to take a photo of in front of a crowd which you are not interested in. You can isolate them from the background using a shallow DoF.