McGonagall sees Harry and Ron loafing around in the hall and suggests that instead of having a free period they might enroll in Potions (classic McGonagall), now that Snape isn't teaching it—you know, since Harry wants to be an Auror (that's a Dark wizard catcher, for those of you just joining us).
So, the boys head over to Slughorn's classroom.
When they get there, class is already underway, and they have to grab the only two remaining textbooks from the cabinet so they can follow along (ah, the hazards of adding a class late). They fight over who gets the only nice, newish-looking one left.
Harry loses (is it just us, or has Ron been working out over the summer?), and ends up with a pretty old and ganky-looking book.
Slughorn demonstrates a few different potions, including one called Felix Felicis, which is basically "liquid luck." If you drink it, you succeed at whatever you're doing. Sounds like something that would be pretty useful around test time, are we right?
Slughorn makes things interesting by offering a vial of it to whoever brews the best potion of the day.
The students start brewing their potions for the prize. Harry opens his tattered textbook and sees that on the first page its previous owner has written "Property of the Half-Blood Prince." and there are tons of handwritten notes on all the pages. Harry uses the "Prince's" helpful margin notes as he works on his recipe.
Remember how pathetic Harry was in Potions up to this point? Well, not anymore—the Prince's notes make him literally the best in the class. He wins the vial.