Snape gave himself the alias of the Half-Blood Prince. The name comes from the fact that his mother's name was Aileen Prince--which means half his blood was "Prince," because it came from his mother, get it?
When Snape was in school, he was highly intelligent and spent a lot of his time making potions better and coming up with spells all by himself. Kids often give themselves cool nicknames, and that's what he did. At the back of his potions book, he wrote, "This book is property of the Half-Blood Prince."
In his potions book, he wrote all his new spells and curses, and he "corrected" the instructions on how to make potions. He found out how to make potions amazingly better than the book originally had them.
Then, Harry needs a book for his new potions class, and so he gets an old one from the closet, that Snape had left in the classroom, and it is Snape's--he uses all the hints Snape wrote in it, and he uses several curses that Snape created, as well (all the while, he had no idea who the Half-Blood Prince was).
The book was included to freak Harry out--when he found out Snape was the Half-Blood Prince (aka, Snape was the guy he had been taking advice from and had been admiring all year), he was forced to realize just how intelligent and powerful Snape really was.
I haven't seen the movie, by the way, just read the book. =D