Matrix-metering evaluates the metering based on a 420 segment sensor which looks at the whole scene, does some sophisticated mumbo-jumbo (magic), and does its best to guess what the best exposure for the whole scene is. It's often, but not always, right. Most professionals agree that 90% of shooting can be done on this mode.
Center-weighted metering looks at the middle of the frame, assumes that that area has an overall average reflectance. (Meaning that if you average all of the tones in that area, you'll get middle gray.) If that's the case, center-weighted metering will give you good results.
Spot metering does the same thing as center-weighted metering, but it does it with only a tiny spot around your focus point. Spot metering works best for night photography since the operator can meter for a specific item of focus rather than blowing out the entire scene.
If you're getting over- or under-exposed shots, your metering system is making inaccurate judgments about the brightness of the scene. This tends to happen when the scene (or the portion being metered by the center/spot mode) contains a predominance of bright or dark tones, so that it fails to average out to middle gray.
By dialing in positive exposure compensation, you will get more exposure, or less, as needed.
AE-L (Auto-exposure lock)
If something in the scene is known to be middle gray, or if there's part of the scene which you want to treat as middle gray for purposes of the exposure, but that part of the scene doesn't happen to be in the middle of the screen, you can use AE lock, then recompose the image, just as with the focus and recompose technique.
The AE lock function can be assigned to either the AE-L/AF-L button above the viewfinder, or to the half-press of the shutter release button. AF lock can also be assigned to either of these buttons, and if you assign one function to each, you can focus on one area of the scene, meter off another area of the scene, but use a different composition entirely to actually take the picture.
Shooting without metering
If you use an older lens without a CPU for metering (meaning manual focus lenses other than AI-P), you'll have to work without metering of any kind.
With practice and knowledge of the Sunny 16 Rule, you may be able to nail shots right off the bat. Until then, you can always take a shot, look at the LCD preview, and adjust your exposure in manual mode. If you can't immediately gauge the needed exposure adjustment, you can also check the three-channel histogram and use that to make a more informed decision about just how you need to alter your exposure.