McClaren, who played for County in the 1980s, was assistant to manager Jim Smith when Derby were promoted to the Premier League before leaving the Rams in 1999 to become assistant to Sir Alex Ferguson at Manchester United.
Clough, who was the longest-serving manager in the Championship, was relieved of his duties following a 1-0 defeat at the City Ground.
McClaren takes charge with Derby 14th in the Championship table and he is aiming to build on the foundations laid down by his predecessor Clough, with promotion back to the Premier League the ultimate target.
He told the FL72 Review show: "We want to build initially on the good work that has gone already. You've seen all the young players that have gone through and that's got to be a credit to Nigel Clough and the staff and the academy.
"They have been bringing young talent through over the last three or four years. The club has stabilised itself after being in the Premier League and coming out, and they've been getting the finances in order.
"What we want to do now is build and develop from that. I'm here for two-and-a-half years initially and we want to assess the situation.
"We want to keep the homegrown players coming through, build on the academy, and build an environment and culture for the players to challenge for a play-off spot, then eventually get in the play-offs and eventually then to be one of the contenders to go up into the Premier League.
"We've got two years to do that. It's a building process and not something that's going to happen overnight."
A statement on the Derby website read: "McClaren, 52, has signed a two-and-half-year deal at Pride Park Stadium and will officially take up his new role at the club on Wednesday, after leaving his position on the coaching staff at Sky Bet Championship leaders Queens Park Rangers.
"He will watch on from the stands for Derby's home game with Ipswich Town on Tuesday evening and will officially take charge of his first match on Saturday (5 October), when the Rams take on Leeds United at Pride Park.
"He will be joined by two other ex-Derby players on what will be a new-look coaching set-up, with Paul Simpson (first-team coach) and Eric Steele (goalkeeping coach) also agreeing to return to the club."
The club also expect to appoint a new technical director in the near future.
President and chief executive Sam Rush said: "We are thrilled to announce Steve McClaren as our new head coach as well as Paul Simpson and Eric Steele in their respective roles.
"Our goal is to have Derby County competing for a play-off place and beyond, and we feel with Steve and his team at the helm we have a great opportunity to do so.
"Steve is a highly qualified coach with all the skill-sets required to be a success here at Pride Park.
"He also has a network of contacts both on a European and global scale, and has also proved in the past that he can identify talent and develop it on the training ground, especially younger players, which is where our Academy comes into play and his working relationship with Darren Wassall is critical to our on-going success.
"It is also important that Steve, Paul and Eric know what it means to play for Derby County and they have been at the club during the good times."
QPR midfielder Joey Barton believes McClaren is well suited to the Derby job and enjoyed his time working with the former England coach.
"He would be a massive loss for any club," said Barton. "I've really enjoyed my time working with him. He's been brilliant.
"I've always thought he's been one of the best English coaches out there and he gets an unfair media because of his England tenure.
"I think he would be a great benefit for any club. He's certainly a very good coach, very organised, puts on great sessions. He's a great fellow and the perfect sort of guy for that job."