Chapter 20 - "The Dementors' Kiss" - In which they go back through the tunnel towards the castle, Black invites Harry to live with him, Lupin becomes a werewolf when the moon comes out and Black transforms to a dog to prevent him from attacking HRH, Pettigrew becomes a rat and escapes, Black goes after him and is confronted by Dementors, Harry and Hermione run for help but are caught by Dementors as well.
Walking through the tunnel back to [Hogwarts], [Sirius Black] offers [Harry] the chance to come live with him, exciting Harry.
The Rescuers Need Rescuing
With almost no warning, [Lupin] begins to transform into a [werewolf]. In the confusion, [Pettigrew] transforms back into a rat and vanishes, and [Sirius] transforms into a dog again to fight the werewolf. But just as [Harry] and [Hermione] prepare to take [Ron] back to the castle, they hear Sirius in pain, and without thinking, run after.
[Harry] and [Hermione] find [Sirius] by the lake, but they also find over a hundred [dementors]. [Harry] tries to cast a [Patronus] and fight them off, but can't manage it, and begins to pass out - but just before he does, the dementors are driven away by something Harry can't quite figure out.
Chapter 21 - "Hermione's Secret" - In which Harry awakens in hospital, desperately tries to tell Black's side of the story but is not believed until Dumbledore arrives, who tells Hermione to use her Time-Turner (which she had been using to increase the amount of classes she could attend) to go back and change events to save Buckbeak and Black. First they rescue Buckbeak, then Harry rescues himself from the Dementors with a Patronus Charm, then they ride Buckbeak to rescue Black, who escapes on the hippogriff.
But Siruis is Innocent!
[Harry] awakens in the [hospital wing] and overhears [Snape] conversing with [Cornelius Fudge] about all that had happened, from his perspective. But when Harry learns that [Sirius Black] is about to be given a [dementor's kiss], he and Hermione beg [Fudge] to reconsider. He doesn't listen, but when [Dumbledore] arrives he asks the rest of the adults to leave - and after some hesitation, they do.
[Dumbledore] admits that he believes [Harry] and [Hermione]'s story about [Sirius Black], but warns them that there isn't enough time to save him. However, Hermione realizes something, and Dumbledore gives her a very mysterious set of instructions before locking them into the [hospital wing].
[Hermione] throws her necklace around [Harry] and twists it, and to his surprise, he is warped to another place. She pulls him into a [broom closet], where she explains that they've gone back in time using her [Time-Turner]. Together, they then realize what they need to do: save [Buckbeak], and fly him to save [Sirius Black].
[Harry] and [Hermione], having travelled back in time, follow themselves down to [Hagrid's hut], hiding behind the trees as their past selves leaves and [Fudge], [Dumbledore], and [Macnair] arrive.
As soon as [Macnair] looks away, [Harry] runs out to grab [Buckbeak]. After a struggle, he and [Hermione] manage to get Buckbeak into the woods, and freeze; from there they overhear [Hagrid] celebrating, Macnair cursing, and, finally, silence.