Simply put, a transmission allows power to be sent from a power source, most often an engine or motor, to a drive mechanism. It uses gears and a clutch to convert the speed of the power source into torque. A simpler one is often referred to as a gearbox since it is basically a box containing a configuration of gears.
The most common example is found in an automobile. There are two types of automobile transmissions: manual and automatic. Both accomplish the same function in turning engine speed or revolutions per minute (rpm) into torque (measured in pounds/feet). They also allow the drive mechanism to shift from forward into reverse without the need to shut off one engine and reverse the direction of the crankcase with a second engine running in the opposite direction.
In the case of a manual transmission, the flywheel, which is connected to the crankcase of the engine, runs at a constant speed. Through the use of a manually operated clutch and a shifter, engine power is reduced and increased significantly through the engagement or disengagement of a series of larger and smaller gears. This shifting of gears will avoid a correspondingly significant, inefficient, and potentially damaging increase of engine speed. These gears are running at different speeds because they are of differing sizes. Larger gears convert higher engine rpm into higher torque or energy at lower drive wheel speed by spinning more slowly than the crankcase; smaller gears, conversely, convert lower engine rpm into higher speed and efficiency by spinning faster than the crankcase.
Automatic transmissions, called transaxles in front wheel drive vehicles, accomplish exactly the same function through speed-torque conversion. A torque converter automatically engages and disengages the correct gears in the proper ratio using a system of bands engaging and disengaging the hydraulically operated clutch pack. This replaces the manually operated clutch. While manual gearing is in line, the gears in an automatic car are in a "planetary" arrangement, meaning they revolve around a stationary "sun" gear.
All of this is enabled through the use of gear reduction where larger gears that rotate more slowly are exchanged via the clutch or the torque converter for smaller, more rapidly rotating gears to increase the speed of the drive wheels. The reverse of the gearing exchange occurs with a need to decrease speed. Therefore, the speed of the power source, the engine or motor, is dampened or enhanced by the manual clutch or the torque converter and clutch pack to increase the efficiency of, as well as decrease the wear and tear on, the engine while providing the needed torque to the drive shaft.
Even such simple power sources as windmill vanes or the pedals on a three speed bicycle utilize a primitive transmission to transfer and partition the energy of the power source rpm into driving torque.