The 49-year-old takes over from Sean O'Driscoll, who was dismissed last week with the Robins languishing in 22nd place in League One.
They have since dropped another place, losing 1-0 to Preston on Saturday, with only Notts County below them.
Cotterill's most recent managerial role was at Nottingham Forest but he was part of Harry Redknapp's coaching staff at QPR on a short-term deal before declining an offer to continue at Loftus Road this season.
Cotterill believes that his new club have the potential to be a Premier League club but that his initial task is to improve their standing in League One.
"Obviously, the first thing I need to do is turn it around," said the 49-year-old. "But I feel Bristol should have a Premier League club. It's a great city and it deserves to have one.
"It has always been my ambition to manage in the Premier League and ambition burns as fiercely as ever.
"There's enormous potential at this club. The long-term plan is in place - that has already been put down by the chairman and the board. For me, in my conversations over the weekend, I think that plan is fantastic.
"A new stadium - a 40,000-seater stadium - there aren't many clubs that can have that kind of vision and I want to be around to be part of that because, as I said before, the potential here is huge.
"But I think in the short term there also needs to be a short-term plan. That's important, and I have one in mind."
Asked if he could turn it around at Ashton Gate, Cotterill said: "If I don't manage to do it then it will be the only one I haven't.
"Nottingham Forest was going down like the Titanic. Forest would have ended up finishing the season with 24 points if I hadn't gone in there. And this is not knocking Steve McClaren, but a former England manager was in charge there before me and I managed to turn that around.
"It wasn't immediate. I went in there around the same sort of time - the transfer window had just closed and it was the back end of November when you couldn't get any players in.
"But I turned it around and I was in charge of a team that went to Leeds and scored seven goals - and no-one had ever done that.
"I want to do it here because I want to do really well for three sets of people. One is the owner and the directors, another is the players because I love my players and I will fight for them tooth and nail, but more importantly it has to be for the fans.
"The fans are here for Bristol City Football Club through thick and thin, and they are everything to a football club and we need to give them something to shout about."