An academy is a place of higher learning, so a dream in which you attend an academy will often mean that you have something to learn.
Your unconscious could be telling you to try to learn from your experiences or to play close attention to the things around so that you can obtain more knowledge from your observations.
An academy is often exclusive, so a dream of being accepted into an academy can reflect hopes of being accepted by society or by a particular group of people in your life.
Dreaming that you are denied acceptance to an academy can mean that you feel that you don’t fit in with others.
A teacher or principal at an academy in a dream can represent authority.
The way you react to that person in your dream can tell you something about your feelings about authority.