N.Y.C. Charter § 6: Section 6. Heads of departments; appoint; remove.

a. The mayor shall appoint the heads of administrations, departments, all commissioners and all other officers not elected by the people, except as otherwise provided by law.

b. The mayor, whenever in his judgment the public interest shall so require, may remove from office any public officer holding office by appointment from a mayor of the city, except officers for whose removal other provision is made by law. No public officer shall hold his office for any specific term, except as otherwise provided by law.