Torque
As the engine turns the propeller one direction, the propeller acts to turn the airplane in the opposite direction, according to Newton's third law.
On most general aviation airplanes, the engine turns the propeller to the right (when the airplane is viewed from behind). As a result, the propeller tries to turn the engine and entire airplane to the left, around the propeller shaft. The airplane is a lot heavier than the propeller, but the engine torque does produce a left turning tendency in the airplane.
Torque varies directly with engine power. It is greatest at high power settings, since the engine is placing right turning force on the propeller.
Spiraling Slipstream
The propeller produces thrust by interacting with the air. As it does so, it grabs the air and throws it rearward. Since the propeller spins to the right, the air it moves is spun to the right, as well. This air spirals around the airplane as it moves rearward.
When this spiraling air strikes the airplane's tail surfaces, it places pressure on the left side of the vertical stabilizer. The airplane has a nose left tendency as a result.
The effects of spiraling slipstream are the greatest at high power and low airspeed. At high power the propeller produces a more forceful spinning force on the air it is moving. At idle power, however, the propeller is effecting the air little. At higher airspeeds, the forward motion of the airplane lessons the effects of the spiraling slipstream. Lower airspeeds allow the spiraling slipstream to be more condensed, circling the airplane in a tighter pattern. This tighter pattern allows the spiraling slipstream to place greater forces on the tail.
Gyroscopic Precession
As the propeller spins, it takes on the properties of a gyroscope. Like a gyroscope, a force applied to the propeller is diverted 90 degrees in the direction of rotation. This is known as gyroscopic precession.
When the pilot pitches the airplane's nose upward, forward force is placed on the bottom of the propeller, while a rearward force is place on the top of the propeller. Since the propeller spins right, these forces are diverted 90 degrees clockwise. The forward force on the bottom of the propeller turns into a forward force on the left side of the propeller, while the rearward force on the top of the propeller becomes a rearward force on the right side of the propeller. As a result, pitching the airplane upward results in right yaw. The opposite is true when the airplane is pitched down, it tends to yaw to the left. Gyroscopic precession is referred to as a left turning tendency in reference to the possible downward pitch of a tailwheel aircraft during takeoff. In a nose wheel aircraft, it does act to place a right yawing force on the airplane during rotation. However, it is still considered a left turning tendency.
Gyroscopic precession is greatest at high engine RPM.
Asymmetric Thrust (P-Factor)
Because the airplane flies with an angle of attack, the relative wind strikes the propeller at a slight angle. As the propeller blades descend on the right side of the propeller shaft, they advance into the relative wind, as a result of this angle. As they rise on the left side of the propeller shaft, they retreat from the relative wind.
The advancing blades on the right sees more airspeed than the retreating blade on the left, causing it to grab more air. The result is more thrust being produced by the right side of the propeller versus the left. This thrust differential pulls the nose of the airplane left.
Asymmetric thrust is most pronounced at high engine power and high angles of attack.
Left Turning Tendencies Acting Together
Torque, spiraling slipstream, and asymmetric thrust are all greatest with high engine power. Spiraling slipstream and asymmetric thrust are both more intense at low airspeeds.
During takeoff and climbout, the left turning tendencies have their greatest effects. The pilot counteracts the left turning tendencies by using right rudder.