Rowling has said that she looked forward to writing Book 3 because it introduced us to Remus Lupin, one of her favorite characters in the whole series (source).
Rowling isn't really a cat person and is allergic to them. However, there was one neighbor cat that she sort of liked in the 1980s, and Crookshanks is based on this cat. She describes this cat as "a large, fluffy ginger cat that looked as though it had run face-first into a wall" (source).
The Prisoner of Azkaban was the last Harry Potter book to be published out of sync in the US and the UK. Readers in the United Kingdom got their hands on Book 3 in July 1999, while their American counterparts had to wait until September. We're still not sure how the American Harry Potter fans managed to survive the wait (source).
For sixteen weeks in 1999, the first three books in the Harry Potter series held the top three slots on the New York Times Bestseller List. Because Harry was clogging up the works there, the New York Times actually created a brand new Bestseller List for Children's Literature (source).
J.K. Rowling has said in multiple interviews that Azkaban was the easiest Harry Potter book for her to write (source).
Guillermo del Toro was offered The Prisoner of Azkaban, but he opted to direct Hellboy instead. So the film was directed by Alfonso Cauron (source).
Sirius's name is a bit of an in-joke: Sirius is the name of a constellation, otherwise known as the "dog star." Sirius Black can transform into a giant black dog in his Animagus form (source).
Odd as it sounds now, some adults in the UK were a bit embarrassed to be caught reading Harry Potter when the books first came out. So UK publishers released two different covers for every Harry Potter book: a kid version and an adult version (source). Go figure. We've got to say that the adult cover of Azkaban is pretty cool, though.