What is a disc brake adapter? Simply put, it's a piece of metal and two screws that allow rotors of different sizes to be mounted on the frame of the bicycle. Basically it is a spacer. Not all disc brakes require adapters: if your bike's frame is designed for 160mm rotors and post mount brakes and you stick to that disc size, an adapter is not needed. This is actually only the case with Post Mount.
Larger diameter rotors dissipate heat faster and therefore perform better under heavy loads. This extra power is more noticeable on the front brake than on the rear. Changing the rear rotor from 160mm to 180mm will not have the same effect as switching to the 180mm rotor at the front.
Adapters are required for larger rotors.