The center-weighted metering determines an average brightness across the whole image but gives more weight to an area in the center.
This center area getting more weight can vary with the camera and some allow even to change its size in menu settings.
This method is the oldest one among the ones I have described here. When the first cameras with automatic exposure metering were developed in the last century they have measured a plain average brightness across the whole subject.
The center-weighted metering was the first refinement of this method before the more complex systems using several sensors came up in the 1980s and then developed further into the complex methods that we see today.