Cooper replaces Chris Hughton at the City Ground
The former Swansea boss, who left the Welsh club at the end of July, replaces Chris Hughton at the City Ground.
Hughton paid the price for a poor start to the Championship season, where they took just one point from their opening seven games.
Forest have had to pay Swansea compensation to get Cooper as he was on gardening leave in south Wales, having just under 12 months left on his contract.
The 41-year-old decided to walk away from the Liberty Stadium after two seasons in charge, where he led them to back-to-back top-six finishes.
Steve Rands and Alan Tate, who were part of his backroom staff at Swansea, are understood to be joining him at Forest.
The trio took training on Tuesday ahead of Saturday’s Championship match with Millwall.
Forest chief executive Dane Murphy told the club’s official website: “Steve was our first choice as head coach and we are delighted to have secured his services. His record of developing young talent is exceptional.
“Steve knows what is required to be successful in the Championship and has a proven track record with Swansea.
“We have reduced the age of our squad over the summer and Steve is the ideal coach to blend a team to start moving us up the table.
“On behalf of the owner and the board, we welcome Steve to the club and look forward to providing every support he needs to be a success at Nottingham Forest.”