The names of the boxes depend on whether you are talking Airbus or Boeing. My experience is on the latter.
The FMC (Flight Management Computer) is the box that calculates the wind. The inputs are:
Track, heading and groundspeed from the nav system (varies between models, but typically inertial with updates from GPS and terrestrial navaids); and
TAS from the ADC (Air Data Computer)
The FMC then does a vector triangle solution, the 3 sides of the triangle being:
Heading & TAS
Track & groundspeed
Wind direction and speed.
On the ones I flew, the displayed data was updated about once or twice a second. Brilliant stuff: going down finals into Palma on a sea-breeze day, for example, you could watch the wind changing from tail to head as you went down the slope. On a lumpy day anywhere, instant and accurate display of the wind affecting you - often much different from that reported by the tower from their distant measuring point.