In a nutshell, the business of exposure is to measure the brightness of the important tones within a scene and translate them into a combination of camera settings that will deliver just the right amount of light to the sensor (or film).
The difficulty is that most average scenes have a brighness ratio - the dynamic range - of several hundred to one, even thousands to one if the sun is in the frame, whereas the sensor can only handle about 128:1 (very rough rule of thumb). That's seven stops - 1, 2, 4, 8, 16, 32, 64, 128 - count the commas. Using Andrew's example above of only five stops, that is reduced to just 32:1, which is pretty tight but not untypical of what you get in a good quality print.
Your eye can adapt to a very wide range of tones instantly and automatically, but because the sensor cannot record them all, you have to choose the important tones, and be prepared to let some others blow to pure white, or at the shadow end just allow them to clog up to solid black with no tonal separation.
An exposure meter attempts to estimate the brightness of the important tones, which are generally around the 18% grey area. It then sticks 18% grey in the middle of the sensor's dynamic range (around the middle of the histogram) and you hope that the other important tones will be contained in the latitude either side.
This pretty hit and miss method generally works okay and is relatively 'safe'. But not always, it doesn't suit all subjects, nor does it usually maximise the sensor's recording potential, some sensors are better than others, and so it goes on etc etc. Working in this way, being guided fairly blindly by the exposure meter, will almost always get you a usable, workable result. But for optimum exposure, you can often get a better result by understanding how it all works and applying that knowledge appropriately according to the subject and the medium (different for different types of film, and for different digital techniques and outputs).
It's a skill, simple as. But the histogram is a fantastic aid. Further, if you know that the important tones cannot be contained within the dynamic range, no matter how carefully you adjust the exposure, then that's when you need to change things further by adding a grad filter, or a polariser, or use HDR technique, or fill-in flash, or a reflector. That sort of thing, to try and get a closer match between the subject's dynamic range and the recording capability of the sensor.
IMHO matrix/evaluative metering is usually the most reliable. This is basically dozens of individual spot meter readings from all over the frame. In this way, the camera not only knows what the average brightness is over the scene (which is pretty much all that other metering methods do) it also knows the dynamic range, where the extreme tones are and how extensive they are, where the important mid-tones are and how dominant they are. It then takes this very valuable information and compares it to a database of thousands of typical images, and comes up with recommended exposure settings. For example, very simplistically, it knows that if there is a lot of bright area along the top of the frame, it is probably sky and so it will add extra exposure to make sure the foreground doesn't go too dark.
Centre-weighted metering just scrambles all the light from the whole frame, with an emphasis towards the centre. Spot metering also scrambles the light into an average tone, but only reads from the centre circle in the viewfinder - use with care!
