The Blades sacked David Weir 12 days ago following a run of just one win in 13 matches.
Clough was dismissed by Derby at the end of September after spending nearly four years at Pride Park. He becomes the Blades' eighth permanent manager in the last six-and-a-half years.
"I am very proud and honoured to be given the opportunity to manage here," Clough said. "I was disappointed with how it ended at Derby but now we have to put that behind us.
"I will try and build a team and squad capable of getting promotion. We are not miracle workers, it takes time to mould and recruit a team.
"I am looking forward to seeing the squad at close quarters, I think things are slowly turning round.
"To be realistic, let's get out of the bottom four and then get a squad together that can challenge."
Clough enjoyed a 24-year playing career which started at Nottingham Forest under his father Brian before he spent time at Liverpool and Manchester City.
He made 14 appearances for England between 1989 and 1993.
In 1998 he took on the role of player-manager at Burton, who were then in the Southern League Premier Division, and took them to the brink of promotion to the Football League for the first time before following in his father's footsteps by becoming Derby manager in January 2009.
Derby escaped relegation from the Championship on the penultimate day of the 2008-09 campaign and finished 14th, 19th, 12th and 10th in the following four seasons under Clough, who left the club in 14th place when he was sacked.
Co-chairman Prince Abdullah bin Mosaad bin Abdulaziz Al Saud said: "Nigel was the clear, first choice of the board. He brings an approach to the game that is well suited to what we are trying to achieve at Sheffield United.
"Nigel has enjoyed managerial success in the game. He knows how to inspire first team players to give their all and has a proven track record of making an academy a key ingredient to the club's success. We count ourselves lucky to have Nigel aboard."
Chris Morgan, who had been interim manager, steps aside with immediate effect although is expected to remain at the club both as Under-21 coach and to assist Clough.
Gary Crosby and Andy Garner, who worked with Clough at Derby and Burton, also join the backroom staff, with Nigel's brother Simon named as a scout.
Clough, 47, is believed to have agreed a severance package with Derby, who sacked him at the end of last month after nearly four years at Pride Park.
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