The 22-year-old midfielder moves to the City Ground on a five-year contract and is Forest's 16th signing of the summer.
Forest have spent most of the transfer window trying to sign Gibbs-White, who came through Wolves' academy.
The Stafford-born player has started both Premier League games for Bruno Lage's side this season.
Forest will pay an initial fee of £25m, plus up to £17m in potential add-ons, for Gibbs-White, who will wear the number 10 shirt.
He scored 12 goals and registered 10 assists in 37 appearances for the Blades, including scoring against Forest at The City Ground in the second leg of the Championship play-off semi-final.