This is the simplest form, where the camera will use all the light coming from the entire scene to determine the exposure setting. No weight is given to any particular portion of the metered area, so an anomalous bright spot, for example, can result in overall under exposure. True average metering is a very rare thing. The vast majority of 35mm SLR film cameras employed the second form of metering.
Centre-weighted average metering
In this system, the meter concentrates between 60 to 80% of its sensitivity towards the central part of the viewfinder. The advantage of this method is that small areas at the edges of the viewfinder that vary greatly in brightness have less influence, and most subjects are generally in the central section of the frame anyway. In truth, centre-weighted metering was more of a consequence that design feature, since light scatter from the focusing screen coupled with the positioning of the meter cell(s) naturally caused an intensity fall off at the edges.
This type of metering works on the same principle as centre-weighted average, but intentionally ignores areas on the edges of the frame, which could otherwise influence the metering unduly if there are either very bright or dark. Partial metering typically concentrates on around 10-15% of the entire frame. Canon was a manufacturer quite keen on this system at one time.
Here the meter will only measure a very small area, typically at the centre of the view screen, and usually between 1-5% of the viewfinder area. Spot metering is very accurate and is not influenced by other areas in the frame. It is commonly used to shoot high contrast scenes. For example, backlit subjects, where perhaps a face is much darker than the bright halo of sunlight around the subject. Spot metering enables the photographer to select which element of a shot is correctly exposed, and the consequential under or over exposure of other, less important areas. Spot metering was usually a second option on high-end cameras, and not an every day metering pattern.
Matrix or Multi-zone metering
This is a much later development, where the camera measures the light intensity at several points in the scene, and then combines the results to find the best compromise exposure setting. Matrix metering was first seen on the Nikon FA, back in 1983. This pioneering camera didn’t sell well, because nobody understood how its metering system worked, and so didn’t trust its accuracy. Yet today, this system is the basis of evaluative metering, the name by which Matrix or Multi-zone has become more commonly known in digital cameras.
Obviously, as we move down this list, the metering systems become potentially more accurate, but the desirability of any system really depends on what you mostly want to photograph. There are no hard and fast rules to say which type of metering is most appropriate, and familiarity with the performance of any metering system is really the best way to exploit its particular strengths and weaknesses.