Basic Physics
The higher air pressure rushing under the wing creates lift, holding the plane in the air, perpendicular to the direction of flight. A side force acts on the fuselage of the plane, perpendicular to the direction of flight, while the plane is banking.
The Ailerons
An aircraft's ailerons are control surfaces on the plane's wing, alongside the plane's flaps. The pilots use the stick in the cockpit to roll one aileron down and the other up; this causes the plane to roll to one side or the other. When one aileron is lifted, the plane banks to the opposite side.
The Rudder
An aircraft's rudder is the vertical-facing flap on the plane's tail. The rudder is controlled by the pilots using pedals in the cockpit. When the rudder is angled to the left, the plane's nose banks slightly to the left, and vice-versa. The motion of the rudder keeps the plane's nose from lifting or falling during banking.