The 50-year-old returns to Parkhead just over four years after leaving the Scottish Premiership giants to take charge of Leicester City.
Rodgers mutually agreed to leave the Foxes in April following a poor run of results which ultimately led to their relegation from the Premier League.
Following positive talks held with Celtic's hierarchy earlier this month, the Northern Irishman has agreed to take the reigns at Celtic for a second time and will succeed Ange Postecoglou, who departed in May to join Tottenham Hotspur.
Celtic have also confirmed that John Kennedy, who was linked with a possible move to join Postecoglou at Spurs, will support Rodgers as his assistant manager, along with first-team coaches Stevie Woods, Gavin Strachan and Harry Kewell, as will B Team coaches Stephen McManus and Darren O'Dea.
Speaking to Celtic's official website on his return to the club, Rodgers said: "I am really delighted to return to Celtic and I am hugely excited by this great opportunity. When I was given the privilege of being asked to join the club again, it was a very simple decision for me and my family.
"We have enjoyed some great times at Celtic before and this will be my goal again, to deliver good football, ensure we have a team we all love to watch and ultimately bring more success to our fans.
"Ange has done some brilliant work across the last two years and I will be doing all I can to maintain Celtic's momentum as we face into all our domestic and European challenges.
"We have a hugely impressive and talented group of players and staff at the club, and I am really looking forward to meeting up with them all and then getting straight down to work."
Rodgers added: "I wanted John to be with me as we move forward. He is a very talented coach, someone I know well and respect and he will be very important to myself and our players.
"The club is in fantastic condition at all levels and I would like to thank the board for putting their trust in me to take the team forward. We will work very closely together as we strive to bring that success to our supporters. Let's get to work!"
During his first stint in charge of Celtic between 2016 and 2019, Rodgers won 69.8% of his 169 matches across all competitions, and lifted two Premiership titles, two Scottish Cups and three League Cups.
Rodgers won back-to-back domestic trebles in his first two seasons at Parkhead, before deciding to leave the club midway through his third season in charge to join Leicester, where he won the FA Cup and Community Shield, and also enjoyed spells in Europe - most notably guiding the Foxes to the Europa Conference League semi-finals in 2021-22.
The former Liverpool and Swansea City boss is tasked with maintaining Celtic's position at the summit of Scottish football and will also hope to improve the club's fortunes in the Champions League group stage in 2023-24.