Easy choice, people.
Honorable mentions go to Harry Potter And The Goblet Of Fire (the most solidly plotted of the series), Harry Potter And The Half-Blood Prince (which misses out on a top spot because Rowling should never join the Skins writing team) and Harry Potter And The Order Of The Phoenix (which is conceptually great, but sometimes fell flat in execution.)
Harry Potter And The Prisoner Of Azkaban is where the series best holds its footing. Whereas its predecessors were clearly children's books with a sprinkle of coming-of-age themes, and its sequels attempted to be fantasy stories that were a magical distortion of the adult world, but ultimately did not quite succeed due to the Potterverse's previous grounding in a children's fantasy land, PoA strikes a balance between a boarding school romp background, dark and adolescent themes (in this case, depression), and a horror-fantasy plot.
Harry has become a more complex and adolescent protagonist, and this theme is present from the start, where, in a fit of anger, he blows up his Aunt Marge and flees the Dursley house. Textually, this is ignored as another comical Dursley incident, but hides something of far greater significance. Harry has committed his first act of violence, however unintentional, and rather than dolefully sitting back and accepting another Dursley punishment, he has outright defied his authority figures and fled out on his own. Harry is rebelling this summer; he has previously partaken in anti-Dursley actions; dealing with Vernon in order to sign his Hogsmeade form, stealing his possessions back from under Vernon's nose. This foreshadows Harry's changing attitude and personality throughout the book, as well as his emerging flaws. Harry is no longer the picture of innocence he was in the prequels; he craftier, more confident, more dangerous.
Indeed, secrets, lies and betrayal, by authority and parental figures, is a common thread throughout the novel. It's in sharp contrast to PS and CoS, where adult witches and wizards, especially those with Harry's best interests, were invariably in the right. In PoA, we see adults make mistakes. Albus Dumbledore, Mr & Mrs Weasley, Hagrid, Remus Lupin, and Sirius Black... all cared for Harry, all wanted to do the right thing, and all of them messed up. It's a huge step forward in Harry's (previously static) character development when he eavesdrops on Hagrid, Professors McGonagall and Flitwick, and Cornelius Fudge at Hogsmeade, and learns about Sirius Black's supposed betrayal of his parents.
And it's here that it becomes apparent that Harry isn't an orphan because some higher power decreed it so. Harry is an orphan because another human being willed it so, and it's a massive slap in the face for our hero. This unlocks a darker streak in Harry's character, one that has been bubbling underneath the surface for sometime now. Harry loses his faith in adults, and goes on a murderous tirade, claiming he will kill Sirius Black, and later tries to make true upon his promise.
The series' growing maturity in terms of thematics is not the only improvement. Prisoner Of Azbakan introduces Sirius Black, Remus Lupin and Peter Pettigrew, some of the most complex characters in the entire series, that are left sadly underused in later volumes. Remus Lupin makes for a wonderful mentor figure, indeed, one more flawed and interesting than Dumbledore. (Although that's for a later Dumbledore-centric essay). Remus is someone who respects Harry on an adult level, unlike previous parental figures such as Mrs Weasley. He is the only teacher Harry seems to trust after the reveal of the fate of Lily and James Potter, the only one that can bring out some exceptional academic prowess, and he even respects Harry's abilities enough to teach him the Patronus Charm. It becomes yet another tragic illustration of the flaws of adult figures when Remus is revealed to be a werewolf, lying to Harry, and seemingly in league with the villain.
I am attempting to limit commentating on Sirius Black, because he fascinates me and I have a later essay planned for him. Sirius does, though, become the most significant in terms of Harry's heroic journey. Harry seems somewhat genre-naive himself, and has constructed an identity for Sirius Black that is one of a monster, a powerful Dark wizard capable of unspeakable crimes. Harry too, constructs a new identity for himself. Harry wants to be the avenging hero, seeking vengeance for the death of his parents. He wants easy closure for his parents murder, something that, frankly, was never given to him.
Except Harry is now past the age where he can play in a world of heroes and villains. When Harry learns that Peter Pettigrew was the true killer of his parents, his worldview is ripped apart. He learns that there was no great evil, no powerful supervillain, just one scared man. Harry understands that there is no point in playing this hero, because, unlike Sirius, whom has fully embraced the role of a tragic hero, he cannot revolve his life around monsters from his past. This character development is solidified when he prevents Sirius and Remus doing what Harry has wanted to do all along, murdering Peter, the enabler of James and Lily's deaths.
Harry rejects his past, rejects the adults shaping his life, and rejects becoming the tragic hero. There is nothing worth hating. Peter wasn't a monster or a supervillain. Peter was a man who chose his own path in life, a path that became one of evil.
Because evil can be human. It's a refreshing change from PS and CoS, where the villain was revealed to have been the omnipresent and unfailingly evil Lord Voldemort, to have the murderer be an ordinary man swept up in a climate of chaos and fear and xenophobia, and it's perhaps the closest that the tragically black-and-white world of Harry Potter comes to creating a villain drawn with shades of grey. There would be no powerful Dark Lord if ordinary and once decent witches and wizards had not decided to follow the Dark Lord. It's a concept that was set up brilliantly in Prisoner Of Azkaban, yet for some reason strangely abandoned in all later volumes.
I can't rave about the villains of Prisoner Of Azkaban without mentioning the Dementors. Dementors rank amongst Rowling's best creations, and one of the few monsters in the series that are legitimately terrifying. A Dementor's gift - removing happy memories and leaving only depression and an empty shell of a person - is a fantastic tie-in for PoA's themes of fear and monsters from the past.
Finally, I could talk for ages on the Harry-Ron-Hermione friendship. Ron and Hermione are perfection in their sidekick roles in this volume, and all three characters play off each other and their roles brilliantly. Harry is the hero, Hermione is the pragmatic realist, and Ron is the honest emotion of the team. This dynamic has been present in earlier books, but it's only here that the friends are forced to confront Harry's judgement, make their own decisions, and in Hermione's case, briefly break away from the gang.
Harry is no longer a child in Prisoner Of Azkaban, but he hasn't yet shouldered the lonely burden he will gain in Order Of The Phoenix. He has seen and learned about the nature of evil, but he has grown greatly as a person, and gained a stronger support network in the forms of Sirius and Remus, and it is these experiences that make him strong enough a hero to triumph in the traumatic events of Goblet Of Fire.
Prisoner Of Azkaban is the novel that made me fall in love with the series. I'm grateful for it.