Forest did not disclose financial details but Sky Sports reported that they had agreed a deal for 12.7 million pounds ($15.45 million) for the 24-year-old, taking their spending during the close season to over 73 million pounds.
Mangala, who has signed a long-term contract with Forest, played 29 games in all competitions for Stuttgart last season and scored one goal. He made his international debut this year and has earned two caps.
Forest have been busy during the summer transfer window, signing former Manchester United midfielder Jesse Lingard (29) on a free transfer.
They have also recruited Dean Henderson, Taiwo Awoniyi, Brandon Aguilera, Neco Williams, Giulian Biancone, Wayne Hennessey, Moussa Niakhate, Omar Richards, Harry Toffolo and Lewis O'Brien.
