Snape is the new Defense Against Dark Arts teacher. The new teacher at Hogwarts, Horace Slughorn, teaches Potions (Snape's old subject).
The Half Blood Prince is a name found in a used Advanced potions book Harry uses that gives helpful potions tips and spells, one of which is extremely dangerous. The Half-Blood Prince turns out to be Severus Snape. Half-Blood refers to the fact that his father was a muggle (Tobias Snape) but the Prince part refers to his mother's maiden name (Eileen Prince).
Harry rightly suspects Draco Malfoy of being a Death Eater. Draco manages to sneak Death Eaters into Hogwarts using a pair of vanishing cabinets. Draco was supposed to kill Dumbledore, but finds himself unable to.
Dumbledore and Harry explore Voldemort's past, learning about Horcruxes, which are divisions of Voldemort's soul, making him immortal until they are all destroyed. There are Seven pieces of his soul (two already destroyed) including the one in Voldemort. Harry and Dumbledore try to get one, but the aftermath leaves Dumbledore weak and unable to fight off the Death Eaters.....
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