The 52-year-old returns to the Championship to take charge of a Robins side that has won just six times this season and is six points adrift of safety.
The club parted company with manager Derek McInnes on Saturday night after 15 months in charge at Ashton Gate on the back of a 4-0 home defeat to Leicester.
And the City board moved quickly to appoint the former Doncaster boss who turned down fellow strugglers Barnsley last week.
O'Driscoll, who has signed a 12-month rolling contract, was relieved of his duties at Forest on Boxing Day after just five months in charge, a decision which shocked many in football.
"We have got to hit the ground running, I haven't got time to assess the squad," he told Press Association Sport.
"I know most of the players from playing against them or seeing them play so I have an opinion of the players as they will have an opinion of me.
"But what is most important in football is the character of the team and I don't know that. I have got to create a little bit of trust quickly.
"My message to the players will be 'show me what you can do.' That doesn't always mean how many goals, how many times you can keep the ball up or how clever you can be. It is more the attitude you show when things don't go your way.
"This is a great opportunity for the players to show me when it hasn't gone their way that they are up for the fight. We've got to do it together."