Lucky for you I just did a speech on digital photography in my Communications class and I have some pictures from my powerpoint.
Pretty much you'll just have to play around with your lens and find out which aperture works best on it. Every lens has it's sweet spot, usually it's somewhere in the middle. With fast primes like a 50mm f/1.8 or 1.4 you'll need to stop them down to ~f/5.6 to get some amazingly sharp pics. Some lenses are designed to perform the best at their biggest aperture (lowest number), these are usually f/2.8 lenses.
Also factor in how much light is available to you and how you'll have to adjust your shutter speed, too low of an aperture (higher number) you'll have to worry about camera shake by extending the exposure.
However as stated with the different apertures you get different depths of field. Here's an example of one extreme to the other.
f/2.8:
f/22:
Another fun thing about the different apertures is how they effect light sources in a picture. Lower apertures (higher number) can give "sparkle"-like effects. These were pretty far in the distance so not as effective, but look at the blue on the extreme left for a good example.
f/22:
