Passive Noise Cancelling headphones are contracted to maximise noise filtering properties. For example, they are packed with layers of high density foam or other absorbing material which can block external sound. This however, makes them heavier than regular headphones, and cannot block too much sound, making them less useful in loud and busy places.
There are many components needed in headphones to achieve this effect:
Microphone - A microphone placed inside the ear cup "listens" to external sounds that cannot be blocked passively.
Noise-canceling circuitry - Electronics, also placed in the ear cup, sense the input from the microphone and generate a "fingerprint" of the noise, noting the frequency and amplitude of the incoming wave. Then they create a new wave that is 180 degrees out of phase with the waves associated with the noise.
Speaker - The "anti-sound" created by the noise-canceling circuitry is fed into the headphones' speakers along with the normal audio; the anti-sound erases the noise by destructive interference, but does not affect the desired sound waves in the normal audio.
Battery - The term "active" refers to the fact that energy must be added to the system to produce the noise-canceling effect. The source of that energy is a rechargeable battery.