For my money, though I like the first two Potter books, this is where Rowling struck gold. I started reading the series in late 1999 or early 2000, well before GOBLET came out, and when I finished the three books that at that time were out, I thought AZKABAN was not only easily the best of three, but one of the best books I had read in a long time. The storyline is easily the strongest of the first three installments, and for once Voldemort is not the main villain driving the plot, but, so it is thought, a renegade supporter of his who murdered 13 people with a single curse.
In AZKABAN, we learn an escaped criminal from the wizard prison Azkaban by the name of Sirius Black is out on the lam looking for Potter. Black was once a vehement supporter for Voldemort, and now Black is gunning to finish off the job by murdering Potter, a task he had tried to do several years ago. Not only that, Potter learns during the course of the plot that Black was James' best friend, along with the new defense against the dark arts teacher, Remus Lupin. We get to learn who Scabbers really is (another instant of an character mentioned in passing on the first two novels who is hugely important here). Black is Potter's godfather, and yet he betrayed the Potters!
What makes Azkaban so interesting is you really get to learn about the relationships between James Potter, Remus Lupin, Sirius Black, Peter Pettigrew, and Severus Snape. These five characters, and their relationships with one another, are huge portions of the foundation on which Rowling built her series. You need a clear understanding of these characters to fully experience Rowling's series, and it is thru these characters that this book, and the series itself, is as rich as it is. The fact no one knew that the three characters were unregistered animagus to help Remus cope with his condition was pretty cool.
For once, Rowling introduces a new magical artifiact called the Marauder's Map, which she uncharacteristically fully explains by the end of the novel. It was made by Padfoot, Moony, Wormtail, and Prongs, which are the nicknames of James and his crew. The map shows you the location of every one on the Hogwarts grounds, a tremendously useful item, supplied, appropriately enough, by those masters of mischief, Fred and George.
Another great new bit of magic in the book is the Patronus, a magical spell that will help fight back the dementors and fear, a very advanced piece of magic for third years. It is also very touching to know why Harry's patronus is a stag, as that is what his father transformed into.
There are also other memorable scenes and events. You get Hermione and the Time Turners, Buckbeak the Hippogriff, Professor Trelawney, the Dementors, the Maurader's Map, etc. The climax of the novel is great, but for me, it's that time when Remus, Sirus, Harry, Hermione, Ron, and Snape are all in that Shreiking Shack, and you finally get to learn a lot of key information about Harry's past.
Ironically enough, though I have long held the opinion this is the best Potter book of them all (not including Book 7), this book has the worst movie adaptation, BECAUSE they don't fully establish all the different relationships between the four, or even explain the Marauder's Map.
For myself, this is easily my favorite of the Potter novels, or was until DEATHLY HALLOWS came out. Still, I have had a great history with this book, and probably reread this more than all the other Potter books. This is the second best Potter book.
These are my order of Potter books by preference:
Prisoner of Azkaban
Order of the Phoenix
Philosopher's Stone/Chamber of Secrets (I rank them both the same)
Goblet of Fire.