The Injustice of Legal Systems
This book makes several moral attacks on a legal system that is controlled by men like Lucius Malfoy who bully people until he gets his way. Due to liability and general xenophobia, Buckbeak is sentenced to execution for harming Malfoy, when every reader saw that Malfoy deserved to be scratched. Furthermore, once Black is caught, only Dumbledore believes that he is innocent, since nobody else cares to listen to a story supported by no evidence other than the words of Hermione and Harry. Cornelius Fudge even says at one point how bad losing track of Black will look for the Ministry of Magic. None of these are fair choices; they are just easy ones. A third choice involving this injustice is the assumption that Crookshanks killed Scabbers. This assumption was supported by evidence. In the cases of this story, the big people are framed, and yet the system won't bother to notice.
The Duality of Life
As shown by Lupin, who spends much of his time as a respectable professor, and then another part as a man-eating werewolf, we understand that everything is capable of having two sides. We see this again when Black is innocent, Hermione begins breaking rules, and Buckbeak's execution is reversed through a simple intrusion through time. Nothing in these stories is ever what it seems; everything stands in a position to surprise. In Harry Potter and the Prisoner of Azkaban, every story has two sides, and in a world where time may change, we have to believe that both of them can be true.
The Importance of Loyalty
The reason Harry feels such personal hatred toward Black is the thought that he betrayed his best friend, James Potter. When it turns out that Pettigrew had done it instead, Lupin and Black turn snarling on him. "YOU SHOULD HAVE DIED!" Black yells at him, "DIED RATHER THAN BETRAY YOUR FRIENDS, AS WE WOULD HAVE DONE FOR YOU!" Harry finds himself facing Black in the first place because he went down the Whomping Willow to rescue Ron. One of the greatest and most repeated messages in this series is summed up by Hagrid's sobering advice to Harry and Ron in chapter fourteen: "I thought you two'd value yer friend more'n broomsticks or rats." Human relationships are the core of this book.