Wilder, 56, signed a contract until the summer of 2025 and replaces Paul Heckingbottom, who was dismissed earlier on Tuesday.
"Quite simply it was an opportunity which I just couldn't turn down," Wilder said on the club's official website. "Just like in 2016, when this club comes calling, it is not something you pass up.
"This is Sheffield United, it is my team and I am thrilled to be back. We find ourselves in a difficult position, I understand that, but I think I can make a difference.
"When I received the call from [chief executive officer] Stephen [Bettis] asking if I would consider coming back to try and help, I didn't have to think about it. People know what this club means to me, and the task now is to try and provide a boost to improve our current situation.
"My relationship with [owner] Prince Abdullah and the board was repaired a long time ago. After some time passed following my initial departure, we spoke, met in person and there was an amicable ending. Now, coming back as manager, to know there is a united front again is crucial as we look to work together to improve the club's position in the Premier League."
Alan Knill, Matt Prestridge and Keith Andrews were also appointed as Wilder's assistants and will all be in the dugout for Wednesday's fixture against Liverpool.
Wilder will hope that his impact this season is as successful as his first spell in charge.
In 2016, his first season at Bramall Lane, Wilder guided the club to the League One title before overseeing their promotion to the Premier League in 2019 as they finished second in the Championship.
An outstanding season in 2019/20 put the Blades in contention to qualify for European football before they ultimately finished in ninth.
The second season did not go as smoothly and Wilder left the club in March 2021 and had spells with Middlesbrough and Watford before his return to the Blades.
The Blades face a tall order to retain their top flight status this season as they currently sit bottom of the table with the most goals conceded and fewest scored.