There are a lot of good replies above Christina,
Why and in what situations (or for what purposes) would you deliberately underexpose a photo using exposure compensation?
How does matrix, center and spot metering come into play in this situation?
Possibly - I think it is important to separate the different issues at play.
I was under the impression from my readings that Center weighted meter is generally used for portrait shots of people and birds, etc, and that spot metering would be used for insects i.e.; macro shots? An oversimplified generalization?
As I see it, the metering method is not directly relevant to getting correct exposure!
To help you separate these two things, let's try an analogy;
Think of two builders; to comply with statutory 'building regulations', they each build houses with a standard ceiling height and a standard light switch height.
One builder always measures the light switch from the floor up, the other from the ceiling down, they are two different measurements, and yet both builders do comply with the regulations, they just approach it from a different direction! It isn't a perfect analogy, but I hope it helps.
A given 'correct' exposure can be arrived by many means;
centre weighted metering
separate ambient light metering
or even guessing
Usually we'll apply an offset to the meter reading, based on experience and/or chimping the histogram and blinkies, using + or - EC, to get the 'correct' exposure.
Which way that is, + or -, depends upon the tones of the subject, background tones and the relative size of the subject in the frame and which metering method we're using.
Which suits me (or you) depends upon the subject, my/our experience of shooting that subject, etc. - but ultimately, it is no more than personal preference for many of us. It is whichever way allows us to quickly mentally re-assess the scene when it changes and apply an appropriate amount of EC to capture it with the 'correct' exposure. Shall we measure up or down to get the switch in the correct place?
... and what is that 'correct' exposure?
Well that depends upon the subject;
if you shoot a scene, say a pale skinned model, and "expose to the right" (ETTR) so the brightest tone in the shot is pushing the blinkies (i.e. you make it 'white' as far as the camera is concerned), you're very likely to need to reduce the exposure in PP to make it look right and that may not even be possible because if you were too far to the right (of the histogram), you'll have gone into the non-linear part of the sensor's photo-receptors.
This is a prime example of when you might say you would be 'under exposing', but you're not really, you are getting the correct exposure. To further complicate things, depending which metering mode you were using, that absolute exposure might have been achieved with either + or - EC on what the meter said.
I will try to shoot some example pictures to demonstrate what I mean, as this is a question that arises quite often and it'll be useful to have some.
I hope that helps, rather than confuses,