The 41-year-old left Northampton Town earlier in the week to take over at Forest after earlier holding talks with Ipswich over a move to Portman Road.
"Forest and Ipswich are similar clubs - they have good youth policies and infrastructures to be better than where they are at the moment," Calderwood told the club's official website.
"The fact Nottingham Forest decided to offer me the job quickly was the thing. I wouldn't allow myself to sit and deliberate.
"This is a fantastic job and a super club and if you did a straw poll of managers and coaches, the vast majority would want to work for Nottingham Forest."