Ghosts are what remains after violent or unexplained deaths. Ghosts did not have to be supernatural beings when they were alive, in fact many were humans so upon arriving at the Academy they were in for yet another surprise. For many ghosts adjusting to being dead is a difficult and sometimes long process. There are only two reasons a ghost will remain around, to try and solve their murder or to rectify a past mistake that is preventing them from moving on.
In the beginning it is very difficult for a ghost to be visible to others, until they get a handle on their power the only time they will be visible is in a time of great distress or anger. But as their abilities grow their power can range from being visible at all times to being able to move things from the other side of the room.