Here's the story:
When Harry got the old potions book from Slughorn, he realized it was once the property of someone called The Half-Blood Prince. The book has instructions scribbled on its pages made by the Half-Blood Prince. Because of the instructions in the book, Harry became a top student in Potions class. Hermione became curious about the personality of the Half-Blood Prince so she did all that she could do to find out who it was.
(In the book)
Hermione had the thought that the Half-Blood Prince was a girl because of the handwriting. Harry did not believe her because he knew the word "Prince" refers to a boy. One day, while in the common room, Hermione showed an old newspaper to Harry which featured a girl named
Eileen Prince. Hermione believed that Eileen Prince was the Half-Blood Prince
Harry ignored her, he had a strong feeling that the Half-Blood Prince was a boy.
Harry and Dumbledore went to a cave and find the locket Horcrux. They returned to the Astronomy Tower. Draco appeared when Harry hid himself under the Invisibilty Cloak. Draco threatened Dumbledore but seemed to be scared. The Death Eaters and then Snape showed up.
Snape killed Dumbledore, Harry chased Snape and used the Half-Blood Prince spells against him.
Snape blocked the spells and was able to disarm Harry. Snape revealed that he was the Half-Blood Prince. Harry was there lying on the ground and couldn't stand the pain, Snape and the Death Eaters escaped.
The next day, Harry talked to Ron and Hermione in the common room. The trio found out that Snape was the Half-Blood Prince, It's because Snape was a Half-Blood (his mother was Eileen Prince, a witch, while his father was Tobias Snape, a muggle) and Prince was his mother's surname.
This part in the book where they found out why Snape's the Half-Blood Prince was not included in the film which is quite disappointing.