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Standard (straight) Wing
Traditional plane designs modelled after popular favorites like the Piper Cub, Cessna 172, Boeing 747, or Sukhoi SU-27, use three types of movable surfaces to twist & bend airflow around to make the plane take off, turn, do cool tricks, and land. In the RC world, these surfaces are controlled by servos connected to linkages, pull cables, and pushrods.
Ailerons are located on the trailing (rearmost) edge of the main wings and are used to make the plane roll, or tilt side to side as viewed from behind -- one wing goes up while the other goes down.
On the miniature wing-like surfaces on the tail of the plane (called horizontal stabilizers) you will find elevators. No, not the kind that take people up & down in tall buildings, but the kind that make airplanes go up & down in the air. Unlike ailerons, elevators operate in unison to push equally up or down on both sides of the tail, thus indirectly turning the nose skyward or towards the ground.
The rudder is a similar device, but there is only one and it is mounted on the upright portion of the tail, called the vertical stabilizer. The rudder is what makes a plane yaw, or turn directly left & right as viewed from above.
In practice, a plane will use all of these control surfaces at once to execute smooth & graceful maneuvers, or to perform the most amazing of aerobatic stunts. Take a simple left turn, for instance. Aileron input will make the plane roll to the left and left rudder will be used to turn the nose. Finally, elevators will pitch it slightly upwards to counteract the loss of lift from the roll and the slight downward turn from using the rudder when tilted to the side.
Some planes use much simpler wing designs that have still very complex dynamics of flight. A delta (isometric triangle -- two equal sides) wing plane has no real tail and instead places hybrid control surfaces called elevons (elevators + ailerons) on the trailing edge of a wing that ends at the rearmost portion of the fuselage. Elevons change the airplane's pitch when used in unison, and cause roll when moved different amounts or opposite of each other. Probably the most famous real-world delta wing aircraft are the F-117A "stealth fighter," the Concorde, and the Space Shuttle.
The least-expensive RC planes you can find use two motors and "differential thrust" to maneuver. To turn, the motor on one side or the other is sped up or slowed down to push the whole plane around. To go up, both motors speed up and more lift is generated, and to come down both motors are slowed. The plane below uses this system -- note the lack of moveable control surfaces on the wings.