Quick start: Set to the default option, which should be titled 'Multi', 'Matrix' or 'Evaluative'.
Your camera's metering system is responsible for choosing the settings designed to give the best exposure for the scene being shot. It decides this by looking at separate parts of the scene and working out which combination of shutter speed and aperture will help to produce the most balanced result.
If shooting in a semi-manual mode such as Aperture Priority or Shutter Priority, the camera will work around the aperture or shutter speed already specified in order to find the other, unspecified setting.
Since most scenes contain a range of tones, from dark shadows to bright highlights, the default evaluative pattern is ideal for general use, and most of the time, on most cameras, it gets it right. Sometimes, however, the evaluative metering system can become confused or give an inappropriate setting, particularly when capturing under awkward lighting conditions such as under bright sunlight.
When the main subject only occupies a small portion of the frame, using the evaluative pattern to balance for the scene as a whole may render the subject too light or dark. In this case, switching to one of the other patterns may be beneficial.
Most cameras offer at least two further patterns: centre-weighted and spot. The former takes the whole scene into consideration, but biases the exposure to the centre of the frame. As such, this is a good option when capturing portraits against a strongly-lit background, one that could easily sway an evaluative system into underexposure of the main subject.
Above: Some cameras have a graphic interface to help you adjust settings such as the metering mode.
Spot metering, meanwhile, uses a much smaller area on which to base exposure (and, in contrast to centre-weighted, only uses this area). As such, it's useful when the subject also only occupies a small portion of the frame, such as a white flower surrounded by plenty of dark green foliage.
Some cameras also go on to offer partial metering, which is similar to spot metering but with a larger measuring area, and also options specifically designed to preserve shadows or highlights. If unsure, and for general use, stick to the default evaluative option.