Did you ever have a slot car? Remember how they had a hand controller that let you change the speed of the car? These are probably the most familiar examples of a DC motor, and illustrates one big advantage, that the speed of rotation can be controlled by changing the voltage. The controller was just a variable resistor. Because the resistor varies the current through the windings, and the strength of the magnetism depends on the current, that changes the speed of rotation.
In a conventional DC motor, the wires are on the rotor, not the stator, and there's an asymmetrical arrangement of coils and magnets. In a small motor like you find in a slot car, there are two magnets and three coils so that they can't line up and lock up (this figure only shows one loop of wire in one coil). Since the coils are rotating, the contacts go through conductive brushes (braided wire or formed carbon) and rings on the rotor shaft connected to the coil wires. (source)