Welcome, Pete - both to APUG and photography.
As noted by others, there are two basic types of meters - incident meters (with the white dome) that are intended to measure the light falling on the subject, and reflective meters that measure the light being reflected from the scene.
Point an incident meter at the light source and it will give you an exposure reading that will render the subject in that light at its "true tonality". Other factors, like nearby reflective surfaces, or multiple light sources, can complicate things, but that's the basic idea.
With reflective meters, the trick is to know what the meter is actually measuring and, thus, what the meter is really telling you. By their nature, reflective meters have a "field of view", and will give you an average of whatever the meter is "seeing" within that field of view. If what the meter is seeing is an even balance of bright and dark areas, the suggested exposure will similarly be a reasonable balance. Otherwise, you have to do a bit of interpreting to get the "right" exposure.
For example, if what the reflective meter is seeing is mostly a white area, you'll want to open up a couple of stops to properly render the scene on film. Conversely, if the meter is seeing mostly shadow areas, you may need to close down a couple of stops.
Spot meters get around this problem to a degree by metering a very small area, and giving you a visual indication of what area they are reading. The result is then easier to interpret, but the same sort of adjustment to the actual exposure will be required.
At the heart of these interpretations or adjustments is the idea that the exposure suggested by the reflective meter is that which will render the metered area as a middle gray - a Zone V, in Zone System terms. Thus, how much you adjust the reading for the actual exposure will shift the metered area up (toward white) or down (toward black) the corresponding amount in relation to the real middle gray in the scene.
The multi-segment matrix meters in modern SLR cameras attempt to make intelligent interpretations of the scene by examining the patterns of brightness and darkness within the camera's firmware. Most of the time, they do a fairly good job, but can be fooled (just like photographers) by unusual lighting situations, such as strong backlighting, etc. Using a good hand-held meter simply puts the photographer in charge of that decision-making process.