Overdrive is a term used to describe the operation of an automobile at sustained speed with reduced engine revolutions per minute (RPM), leading to better fuel consumption, lower noise and lower wear. Use of the term is confused, as it is applied to several different, but related, meanings.
The most fundamental meaning is that of an overall gear ratio between engine and wheels, such that the car is over-geared, and cannot reach its potential top speed, i.e. the car could travel faster if it were in a lower gear, with the engine turning at higher RPM.
The purpose of such a gear may not be immediately obvious. The power produced by an engine increases with the engine's RPM to a maximum, then falls away. The point of maximum power is somewhat lower than the absolute maximum RPM to which the engine is limited, the "redline" RPM. A car's speed is limited by the power required to drive it against air resistance, which increases with speed. At the maximum possible speed, the engine is running at its point of maximum power, or power peak, and the car is traveling at the speed where air resistance equals that maximum power. There is therefore one specific gear ratio at which the car can achieve its maximum speed: the one that matches that engine speed with that travel speed. At travel speeds below this maximum, there is a range of gear ratios that can match engine power to air resistance, and the most fuel efficient is the one that results in the lowest engine speed. Therefore, a car needs one gearing to reach maximum speed but another to reach maximum fuel efficiency at a lower speed.
With the early development of cars and the almost universal rear-wheel drive layout, the final drive (i.e. rear axle) ratio for fast cars was chosen to give the ratio for maximum speed. The gearbox was designed so that, for efficiency, the fastest ratio would be a "direct-drive" or "straight-through" 1:1 ratio, avoiding frictional losses in the gears. Achieving an overdriven ratio for cruising thus required a gearbox ratio even higher than this, i.e. the gearbox output shaft rotating faster than the engine. The propeller shaft linking gearbox and rear axle is thus overdriven, and a transmission capable of doing this became termed an "overdrive" transmission.