The two clubs agreed a fee for Fraser on Tuesday and the 18-year-old has signed a three-and-a-half-year deal at the Goldsands Stadium.
"He is a talented lad," Bournemouth manager Eddie Howe told the club's official website.
"He's only 18 but he's very, very highly regarded in the world of football, especially up North and Aberdeen were desperate not to lose him.
"He came to our attention a while back and he's someone that we have tracked for a long period of time. We had a lot of competition for his signature but the fact we were able to sign him is a real positive for his football club."
Fraser made 18 appearances for Aberdeen this season.