Collins becomes new Manager O'Driscoll's second Reds signing after the club paid an undisclosed fee for his services.
He follows the arrival of Adlene Guedioura from Wolves earlier in the week.
Collins flew back from Stoke's pre-season tour of the US to complete the formalities of his switch to The City Ground.
Nottingham Forest Chairman Omar Al Hasawi said: "We are delighted to bring a player of Danny's experience and calibre to the Club.
"We are sure he will prove to be an excellent addition to the squad Sean O'Driscoll is assembling for the challenges that lie ahead."
The 31 year old Welsh international, who can operate either in the centre of defence or at left back, started his career with Chester City before moving to Sunderland in 2004.
He made more than 150 appearances for the Wearside club, many of which were in the Premier League.
Collins joined Stoke in 2009 and has made 50 top flight appearances for The Potters. He also had loan spells with Ipswich - he scored twice against Forest in a 3-2 defeat at The City Ground - and West Ham last season.
He has been capped 12 times by Wales.