There are basically 2 types of Active Noise Cancelling headphones available on the market - Feedback and Feed-forward, both of which are comprised of three inter-related parts:

A microphone that measures incoming noise and feeds that information to a computer or processor circuit.

The processor that converts the noise information into anti-noise instructions.

A speaker that is driven by the anti-noise signal to generate sound waves that are exactly 180 degrees out of phase with the unwanted signal and of the same magnitude.