Why are the armature and field windings of an electric motor usually wound on an iron core?

Answer 1

Because iron is a ferromagnetic material which can have a higher density of magnetic moments lined up in the same direction making a stronger electromagnet.

Iron, as a magnet, has good Currie temperature, about 1050 F, suitable for industrial use. Above Currie temperature a material turns from ferromagnetic to paramagnetic causing the magnetic moments harder to line up in large fraction.

Fully annealed soft iron also has lower density of grain boundary and carbon atoms in interstitial dislocations as compared to work-hardened iron or carbon steel. This gives lower hysteresis loss in AC applications.

Answer 2

Armature and windings are usually wound on a soft iron core so that more magnetic lines of force may pass through soft iron,resulting in more magnetic flux, and therefore more torque according to formula.
F=B.I.L