Harry uses the Felix Felicis potion to gain Slughorn's memory of telling Tom Riddle/Voldemort about Horcruxes, which are magical items that one can store his soul in after taking a life. Tom expresses interest in creating seven for himself. One was his diary, destroyed in Book/movie 2 (The Chamber of Secrets), another is a ring that Dumbledore gained offscreen before the beginning of the movie (that's why his hand is charred). Dumbledore tells Harry of another Horcrux he knows of, located in a cave isolated from the rest of the world. Harry and Dumbledore retrieve the locket and return to Hogwarts. Here, they are confronted by Draco Malfoy, revealed to be a newly initiated Death Eater, who has been trying to kill Dumbledore on orders from Voldemort.
Bellatrix Lestrange, Narcissa Malfoy, and two other death eaters show up (one is supposed to be Fenrir Greyback, but his face is never shown). Draco is unable to cast the killing curse, which gives Snape the opportunity to arrive. Dumbledore looks at Snape and says, "Severus, please," so Snape casts the killing curse and Dumbledore falls to the ground below. Enraged, Harry follows Snape and begins casting spells he found in his potions book, which is revealed to have been Snape's, as he is the Half-Blood Prince. The Malfoys, Snape, and the Death Eaters escape Hogwarts.
It is revealed that the supposed Horcrux Harry and Dumbledore found in the cave was a fake, with a letter from R.A.B. promising to destroy the real Horcrux before he can be found. Harry, Ron, and Hermione plan to destroy the five remaining Horcruxes in the following two movies.
I'll give away a little bit right now. R.A.B is Regulus Black, Sirius Black's brother, a former Death Eater. He is not featured in any of the stories except in reference.