Now, to get confusing, the way that the numbers work when referring to the f-stop is the complete opposite of what you think. To allow more light in your image, you want the aperture to be more open, the more light you want, the lower the f-stop number will be, for example from f8 down to f3.5.
Practicality!
When you use a low f-stop, you allow more light into your picture (open aperture= faster shutter= no blur/freeze the image). But due to some phenomena unknown to me, comes the term "depth of field" or DOF for all you cool kids. DOF refers to the area in the photo that will be in focus (nice, clear and sharp). When you take a photo with a low f-stop (e.g. f1.8) or really open aperture, the DOF is smaller than when you use a high f-stop (e.g. f16) or small aperture.
So, if you wanted to take a photo where an object, whether a flower or a person, is the main part of the photograph then a low f-stop will make sure that the object is in focus and the areas around it will be nice and blurry so that if someone looks at it, their eyes will be drawn to the subject and not distracted by a busy background.
If you were going to take a landscape shot, then you would want to make sure that the foreground, the middle ground and the background were all sharp and in focus, therefore i would use a high f-stop, where the DOF would be greater.
On standard lenses you usually get an aperture of around f3.5-5.6. You will usually pay more for the lower f-stop numbers from f-2.8 and lower, but they are usually very nice lenses with very good glass in them.
Once again there is a lot more to this than all I have written, but it would be best to be discussed! So please comment and let us know your own views!
Peace
Andy