Exposure is a complex topic.

In order to get the correct exposure, we could spend a lot of time analyzing the quantity of light in a scene.  But when all is said and done, the most complex analysis still may not yield the "best" exposure.  It's a very subjective and elusive thing - this "best" exposure.  You could capture all of the detail in a scene but still not get the "best" exposure.  Often, you won't know what the best exposure is until AFTER you get your pictures back - a bit late?

What constitutes the "best" exposure??  Many things.  Things like the ability to capture details, color saturation, overall density, etc.  You may want to underexpose an image to give it a dark-foreboding look or to achieve better color saturation.  You may want to overexpose an image to wash out highlights you don't want.  Often, you won't know which image you like better until you actually SEE it. 

One technique that helps assure that you will get the "best" exposure is to "bracket" your exposures.  This means you are probably going to take either 3 or 5 pictures of the same scene.  Each of these exposures is going to be at a different exposure.  Let's say, your camera meter indicates that the correct exposure is 1/125 at f8.  You might take three pictures - each one with a single f-stop difference by varying the aperture.  For example, these exposures would give you one picture taken at what the camera recommends, one underexposed by one stop and one overexposed by one stop:

 

 

You could also accomplish the same result by varying the shutter:

 

Which way you choose to bracket is up to you.  The effect on the exposure is the same whether you vary the aperture or shutter, but the depth of field is modified when you vary the aperture and the way motion is handled varies when you modify the shutter speed.

You don't have to vary each exposure one full stop, you can vary it by more or less, depending upon what you're trying to accomplish.  Most electronic  cameras allow you to vary your f-stop by 1/3 or 2/3 thus you could take 3-exposures:  One per your meter, one 2/3 of a f-stop underexposed and one 2/3 of a f-stop overexposed.  This is actually what I do.  I find 2/3 of a stop in either direction works very well with the film I most often shoot:  Fuji Velvia.

Why do this??
You might be asking yourself,

"Why waste film like this??  Is the exposure REALLY that critical??"

Well, it does USE film - but does it waste it??  Depends upon your perspective.  You WILL see a difference in all three exposures and you will most often like the "middle" exposure.  I find that about 70% of the time I like the middle exposure best, but about 30% of the time I like the over or under exposure best.  Also, sometimes, my meter is fooled and if I didn't catch it, I will find that two of the exposures are just plain bad and one is salvageable. 

When should I do this??
You shouldn't have to bracket ALL of your pictures, but you should consider it when you're presented with: