Your question is a good one. An electric motor relies on electrical current, either direct or alternating, that flows through a coil of wire. This coil of wire is between the north and south poles of a magnet. In general, when current flows through a wire in the presence of a magnet, a force is exerted on the wire. This force is named the Lorentz force or Laplace force after the men who described it. An electrical motor is designed so that this coil of wire turns perpetually in place as long as electrical current runs through it. An axle is fixed on the same component on which the wire coil is mounted; as a result, when the current turns, the axle turns too.

There are a lot of neat little details about how electric motors work. In short, it's the presence of a magnet in an electrical motor that functions as a key ingredient in getting movement out of electricity.

If you've got access to the internet, here is a cool application that illustrates what I'm talking about. Just hook up the leads, turn on the switch, and crank the voltage! The checkboxes in the lower left toggle the display of the directions of the magnetic field, electrical current, and Laplace force.