Servos have small electric motors in them that provide power. The motor has a rotor with two attached to it, one on either side. There are two magnets outside of the motor, also on either side. When current flows through both coils, a magnetic field which turns the rotor so that they face the magnets. When the coils pass their magnets, they get another pulse of electricity, pushing them towards the other magnet. The rotor gets two strong pushes every rotation, which keep it turning. The motor is attached by gears to the control wheel.
The control wheel is where the servo motor attaches to whatever control mechanism it is powering. It usually can turn 90 degrees in either direction from the starting position for a total of 180 degrees of movement. The control wheel uses a potentiometer to tell exactly how far it has turned and when to stop. The potentiometer is a variable resistor, a device which creates electricity. As the motor turns, the potentiometer creates more or less resistance. By measuring the potentiometer, the servo controller can tell how far it has turned and send a signal to tell it when to stop.
Controlling the Servo
Servo motors usually have three wires, two which provide power and one which provides a control signal. The control signal uses pulse width modulation. It sends pulses of electricity to the servo every 20 milliseconds. The duration, or "width", of each pulse tells the motor what position to turn to. The motor has control circuitry inside that sends signals to move into position and to stop once it gets there.