Originally Posted by LunaZ

In a nutshell, "metering" is measuring how much light is available in the shot you're seeing through the view finder, which then determines how you should set your shutter speed, aperture and ISO relative to each other to achieve a proper exposure.
The different metering modes tell the camera where to measure the light to guide you to correct exposure. For example you might want to consider the whole frame or just one single spot in the view finder. In the case of of center-weighted metering modes, you're giving priority to the center area of the view finder while still taking the outer edges into consideration.