Which part of the guitar vibrates?

Answer 1

Well the strings do the vibrating. But those vibrations are transferred through the saddle and bridge (the parts that connect the strings to the body) into the body wood which creates the acoustics and amplify the sound. It is then expelled through the sound hole in the front.

If you're using an Electric Guitar the pickups directly under each string will pickup the sounds and transform that into an electrical signal that sounds like a guitar or whatever effect you're using.
All of it, but principally the strings and the box (which responds in resonance with the vibrations of the strings).

Answer 2

In an acoustic guitar the string cause vibrations in the air that reverberate in the hollow body of the guitar. In an electric guitar the pickups create a magnetic arc and when the strings vibrate they pass the magnetic arc, the speed of vibration (frequency) is what creates the pitch that comes out of the amp.