Cotterill, who had been in charge at Portsmouth since June last year, has signed a three-and-a-half-year contract with the Championship club.
"I'm very excited - it's a wonderful football club which has great traditions which should be embraced," Cotterill said on the club's website.
"I'm really looking forward to getting to work on the training pitch with the players."
Forest, who won back-to-back European Cups in 1979 and 1980 but are now just above the Championship relegation zone, parted company with their former England coach Steve McClaren 12 days ago after a dismal start to the season.
Cotterill, who leaves Portsmouth with the south-coast club also struggling in the bottom half of the table, added: "It's been a sad day for me.
"It was very emotional leaving Portsmouth because of the relationship I've had with the staff and players down there over the last 17 months."
Cotterill will begin work on Monday. His appointment comes two days after former Forest player and manager Frank Clark was appointed chairman of the club.
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