Absorbing the power
A propeller must be matched to its engine: it has to convert (absorb) the full power without overspeeding. The design characteristics that increase power absorption are the ones that give the blades more work to do: a larger diameter, more blades, a wider blade chord, a higher rpm, and more blade angle and camber. The obvious first choice, diameter, runs into hard restrictions:
Tip speed — it grows with radius, and a tip running into compressibility loses efficiency and gets loud.
Ground (and fuselage) clearance — it sets a geometric ceiling.
Weight and gyroscopic loads — a bigger disc means more weight and stronger gyroscopic loads.
Once the diameter is capped, the remaining levers carry the growth.
The most important remaining lever is captured in one word, solidity: the ratio of the total blade area to the propeller disc area. Raising solidity by adding blades is the transport answer, and the syllabus asks for both sides of it. Advantages: more power absorbed at the same diameter and tip speed, smoother running, and less noise (each blade more lightly loaded, and no need for the tip speeds a two-blade absorber would demand). Disadvantages: each blade flies in its neighbours' disturbed air, so efficiency per blade falls, and the propeller grows heavier, costlier and more complex. Noise, finally, is minimised by the same family of choices: lower tip speed (rpm and diameter held down), more blades sharing the load, and modern swept, scimitar-shaped tips that soften the tip's compressibility — plus, on multi-engine aircraft, synchronising the propellers so their beats do not add.
Torque reaction
Now the four secondary effects, each a moment the pilot never asked for. The first is bookkeeping the course has done since Newton: the engine twists the propeller one way, so the propeller twists the aircraft the other. Torque reaction rolls the aircraft opposite to the propeller's rotation: a clockwise propeller (seen from behind) rolls the aircraft left. In flight the ailerons hold it easily; on the take-off run it works differently: the roll reaction presses one main wheel harder onto the runway, and the extra rolling friction on that side yaws the aircraft — for the clockwise propeller, to the left. The design answers the syllabus names:
counter-rotating propellers — a twin's two propellers turning in opposite directions, cancelling each other's torque across the aircraft (and, as the critical-engine section shows, abolishing the critical engine).
contra-rotating propellers — two propellers on the same axis turning opposite ways, cancelling torque and untwisting the slipstream on a single engine.
Gyroscopic precession
A spinning propeller disc is a gyroscope, and gyroscopes answer forces strangely: a force applied to the rim acts as if applied 90° later in the direction of rotation. That is gyroscopic precession, and its cause is nothing mystical: each blade's momentum carries the applied push a quarter-turn onward before it has its full effect. For the aircraft the consequence is a cross-coupling of pitch and yaw: pitch the nose and the propeller answers with yaw; yaw the nose and it answers with pitch. For the clockwise propeller (from behind): pitching down yaws the nose left; pitching up yaws it right — felt most sharply when the tail comes up on a tailwheel aircraft's take-off roll (pitch down: yaw left), or at a brisk rotation (pitch up: yaw right).
Slipstream and P-factor
The third effect rides in the propeller's wake. The slipstream leaves the disc rotating, and the helix wraps around the fuselage on its way aft, striking the fin on one side. For the clockwise propeller the spiral comes around to press on the fin's left surface: a yaw to the left, strongest exactly when the slipstream is strongest and the airspeed lowest: full power, early take-off roll. (Designers answer with an offset fin or engine line; pilots answer with rudder.) The slipstream has quieter effects too: the washed parts of the wing and tail fly in faster air, so they lift more and the elevator and rudder answer more crisply at low IAS; every power change re-sizes those contributions, which is part of why configuration and power changes need retrimming. On a twin with an engine out, only the live engine's wing is still washed: the 'asymmetric slipstream' rolling moment that the control module's engine-out aileron work balances. The fourth is the asymmetric blade effect (P-factor). Fly nose-high and the propeller disc is tilted to the oncoming flow: the down-going blade meets the air at a higher effective angle of attack (and slightly faster) than the up-going blade, so the disc's thrust centre shifts toward the down-going side — for the clockwise propeller, to the right of the shaft, yawing the aircraft left. P-factor grows with power and with α: negligible in cruise, prominent nose-high at full power.