The first large family of magnetic actuators is based on the magnetic forces acting at distance, the Laplace-Lorentz forces and the reluctance forces. Linear types offer strokes in the range of 1 to 20mm, and are complementary to piezo actuators. They can be sorted in 3 categories:

Moving coil actuator : Placed in static magnetic field, a mobile coil driven by a current is submitted to the Laplace-Lorentz force. This force is proportional to the applied current. Thus these actuators are controllable. Since first applications were loud-speakers, they are also called voice-coils. 
Moving magnet actuator : Placed between two magnet poles, a mobile permanent magnet can be switched from one pole to the other using coils. Such moving magnet actuators are bi-stable. They present high forces but are not so controllable. 
Moving iron actuator : A soft magnetic part placed into a coil system naturally moves in a way that minimises the system magnetic energy. In standard electro-magnets, this reluctance force is larger than the Laplace force but it is only attractive and not controllable.

The second family of magnetic actuators is based on magnetically-controlled active materials. 

Magnetostrictive actuator : These actuators use the magnetostrictive effect, which consists in the deformation of a magnetostrictive material when subjected to a magnetic effect. For example magnetostrictive deformation of Terfenol-D achieves 1600ppm, making this active material an interesting competitor of piezoelectric PZT ceramics for high power transducers or low voltage actuators.
Magneto Rheological Fluid Actuator : A Magneto Rheological Fluid (MRF) can solidify when subjected to a magnetic field. This effect can be used to design various MRF-based actuators : valves, brakes, clutches, semi-active dampers, smart hydrostatic / hydrodynamic bearings…
The third broad family of magnetic actuators is the electric motors also called electric rotating machines.