The Half-Blood Prince's potion book that Harry uses in Book 6 belonged to Severus Snape, who created the handwritten annotations and spells as a student. Snape was the Half-Blood Prince (his mother was a Prince, his father a muggle), which Harry discovers when he confronts Snape during the climax of the book. This is a key plot point that reveals Snape's dual identity and his potions mastery.