The term "hang" means that you do the exercise from a hang position and not from the floor.
The term "power" is for doing the exercise without the squat.
Hang cleans=from hang position and with squat
Hang power cleans=from hang position without the squat
cleans=from the floor with squat
power cleans=from the floor without the squat
Without the squat means, that you do only a quarter squat to receive the bar.
You can also notify from which exactly position is your hang clean.
For example hang clean from knees, or above knees etc.
And if you use a rack to pull etc, then you can notify as rack power cleans etc.
This is a good site with all the olympic lifts and the drills of them, with videos where you can find all the terms, positions etc
Catalyst Athletics: Resources