The links provided by gryphonslair99 are pretty good at explaining it.
It's really not difficult once you read those. I assimilated metering modes as the area of the entire frame the camera is trying to expose for.
Spot - means the exposure is going to get set so that small portion of the frame will come out properly exposed...and the camera ignores everything else around it.
Center weighted - the camera will consider exposure for a larger area of the frame but will give the most consideration to the center area.
Evaluative (Canon), or Matrix (Nikon) - the camera tries to see everything in the frame and adjust exposure so it all comes out.
Now, that doesn't necessarily tell you what your photo is going to look like. I found that setting up a shot, then taking photos of it and changing the metering mode really told me the most about the effect. I'm a kinesthetic learner, so I tend to pick up things faster by doing than by reading....that's not to say I wouldn't read about it first, but doing it gives me the most feedback and I learn most from that.
As for when you would use each, that depends on what you want your photo to end up like. Landscape might favor matrix because you typically are trying to get the entire frame exposed correctly. Situations like maybe a car shot might favor center weighted...you are trying to get a smaller area than the entire frame correct to emphasize the car. Spot is very useful, but typically you will use it in situations that have large variances in light within the frame and you want to be sure to get the subject right and ignore the rest. The face of a model, for example, or someone's face when they are back lit, or if you are shooting into the sun and want a small subject exposed correctly. That doesn't mean that any time you are shooting into the sun you must use spot, you might decide you want a silhouette instead. use matrix then so the bright sun isn't blown out, this in turn will make the subject way under exposed..i.e. silhouetted.
If you are shooting still life, aren't in a hurry or don't need to get it right the first shot, you can even shoot aperture, or shutter pri, or manual and just take several shots making adjustments in between to get the desired effect too. That's the beauty of digital, shoot away like a mad man and delete what you don't like.