The Villans paid £16 million for Cash, who signed a five-year contract, according to BBC Sport.
Cash joins the club after being named Forest's Player of the Year following a superb 2019-20 Championship campaign at right-back. Head coach Sabri Lamouchi moved Cash into the position after he emerged from the side's youth academy as an energetic midfielder or winger.
He scored three goals and assisted seven times last season. Cash often kept former Arsenal full-back Carl Jenkinson out of the lineup as he boosted his reputation through his understanding with Joe Lolley down the right flank and how quickly he developed the defensive side of his game.
"We are delighted to welcome Matty to Villa and believe we have secured one of England's brightest young prospects," Villa manager Dean Smith said. "He has an excellent playing record for a young player and we believe he will settle in easily and continue his rapid development into a top full-back."