Professor: Care of Magical Creatures
Hagrid has kind of a rough year in this book. He becomes the new Care of Magical Creatures teacher, which really should be the perfect job for him. Unfortunately, Hagrid's first class is a disaster; one of his hippogriffs, Buckbeak, attacks none other than Draco Malfoy. Granted, Draco provoked the hippogriff, but the damage was done. Malfoy pitched a fit, the school's Board of Governors got involved, and next thing we know, Hagrid is embroiled in a months-long legal struggle. Buckbeak loses his case and ends up sentenced to death.
With all this, Hagrid grows steadily more depressed over the year (we even see him drunk a few times). And his teaching suffers as a result – his class becomes horribly boring after Hagrid loses his confidence. But things start looking up for Hagrid by the end of the year. Buckbeak fortunately escapes his executioner thanks to a time-traveling Harry and Hermione, and a joyful Hagrid celebrates by crying and drinking. Which is what he does when he's depressed too, come to think of it.
Aside from dealing with legal problems, Hagrid plays a big role in this novel's theme of friendship. He and Hermione really seem to bond while they're working together on Buckbeak's case, and Hagrid even takes Harry and Ron to task for their petty fight with Hermione, which went on far too long in Hagrid's (and our) opinion.