"I am honored to have this opportunity with such a first-class organization and would like to thank [the Nets front office] for having faith in my ability to lead this team forward," Nash said in a news release.
"Coaching is something I knew I wanted to pursue when the time was right, and I am humbled to be able to work with the outstanding group of players and staff we have here in Brooklyn.
"I am as excited about the prospects of the team on the court as I am about moving to Brooklyn with my family and becoming impactful members of this community."
The Nets finished 35-37 during the NBA regular season before they were swept by the Toronto Raptors in the first-round of the playoffs in Orlando, Fla.
Brooklyn said Vaughn will revert to his role as lead assistant. Nash's contract is for four years.
Nash, 46, was an eight-time All-Star and two-time MVP during his 18-year NBA tenure from 1996 through 2013. He was inducted into the Naismith Memorial Basketball Hall of Fame in 2018.
"In Steve we see a leader, communicator and mentor who will garner the respect of our players," Nets general manager Sean Marks said. "I have had the privilege to know Steve for many years. One of the great on-court leaders in our game, I have witnessed first hand his basketball acumen and selfless approach to prioritize team success.
"His instincts for the game, combined with an inherent ability to communicate with and unite players towards a common goal, will prepare us to compete at the highest levels of the league."