how does the rudder work?
When the rudder is deflected, it creates a "lift" on the vertical tail due to additional camber. This "lift", when viewed from the airplane perspective, is a side force opposite to the direction of the rudder deflection. Since the centre of pressure of this side force is aft of the centre of gravity, it generates a net yaw moment on the airplane.


  For example, a right rudder deflection generates a left side force, which generates a nose right yawing moment, pulling the aircraft nose right.


Since the side force is relatively small in the big scheme of things, the airplane motion is essentially unchanged. As soon as the airplane nose begins to yaw, the airplane begins to have finite sideslip against the incoming airflow. This sideslip generates a side force that opposes the rudder force (from an increased airflow incidence against the vertical tail), as well as a yawing moment that opposes the yaw moment from rudder (i.e. a restoring moment).


  For example, a right rudder generates nose right yaw motion, which creates a sideslip that produces a nose left yaw moment.


For a finite rudder deflection, the airplane begins to accumulate sideslip, until such a point that the restoring moment balances out the rudder moment.