By using magnetorheological (MR) fluid in place of lubricating oil in a traditional squeeze film damper (SFD), one can build a variable-damping SFD, thereby controlling the vibration of a rotor by controlling the magnetic field. Assuming a Bingham model, the Reynolds equation for an MR fluid squeeze film is developed and solved to provide expressions for the velocity, the pressure distribution and the damping force. Electromagnetic theory is used to calculate the magnetic pull force between the magnetic poles in the damper. The mechanical properties of the squeeze film and the unbalance response characteristics of an MR fluid SFD–rigid rotor system are analyzed theoretically. An MR fluid SFD is designed and manufactured, and the unbalance response properties and control method of a flexible rotor supported on the damper are studied experimentally. The study shows that the magnetic pull force can decrease both the first critical speed and the critical amplitude; the film damping force can decrease the amplitude at the undamped critical speeds, but increase the amplitude in a speed range between two undamped critical speeds. The damper may have the best control effect to minimize the vibration within the range of all working speed by using the on–off control method.