With a Lightning a USB-C headphone you plug the headphone directly into the Lightning port (on Apple devices) or USB-C port (on Android devices). A standard headphone plug is an analog connection while this creates a direct digital connection. The headphones are powered by your phone (they use only a little bit of battery power) and have an integrated DAC (digital-to-analog converter) that's usually superior to the DAC in your phone.
The impedance of headphones is of concern because of the output limitations of amplifiers. A modern pair of headphones is driven by an amplifier, with lower impedance headphones presenting a larger load. Amplifiers are not ideal; they also have some output impedance that limits the amount of power they can provide. To ensure an even frequency response, adequate damping factor, and undistorted sound, an amplifier should have an output impedance less than 1/8 that of the headphones it is driving (and ideally, as low as possible). If output impedance is large compared to the impedance of the headphones, significantly higher distortion is present. Therefore, lower impedance headphones tend to be louder and more efficient, but also demand a more capable amplifier. Higher impedance headphones are more tolerant of amplifier limitations, but produce less volume for a given output level.