Most small electric motors in Britain spin at 1425 rpm, whilst those in the USA and Europe are usually marked a little faster at 1600 to 1700 rpm or so.
If the lathe spindle was to be driven directly from one of these motors, even using a small pulley on the motor shaft, and a larger one on the lathe, it would be turning far too fast to be useful for the majority of jobs. Hence, it is necessary to introduce some way of reducing the speed - and that is the job of the countershaft. In a typical arrangement, illustrated below, the motor is fastened to a vertical cast iron plate, hinged at it base, and fitted with a small pulley on its spindle. Because the 1500 rpm motor is driving a much larger pulley above it in a ratio of something like 5 : 1 - the speed of the upper pulley is reduced to 300 rpm (1500 divided by 5).
On the same shaft as the large pulley is a set of three pulleys, usually identical to those on the lathe, but arranged in the "reverse" order. If the middle pulley on the countershaft is made to drive the identically-sized pulley on the lathe spindle that too, of course, will turn at 300 rpm. The pulleys each side of the centre are normally arranged to halve and double the speeds - hence the creation of speed set covering a useful 150 rpm, 300 rpm and 600 rpm.
It is a simple matter to fit both a small and a larger pulley on the motor shaft, and two correspondingly larger pulleys on the countershaft, and so double the number of available speeds - and then to replace the three-step pulley with a four-step - so creating (with a backgear) a sixteen speed drive that, typically, would give a range starting at 25 and extending all the way up to a little over 2000 rpm.
One question that crops up frequently is, "I don't have a pulley on my motor. How big should it be ?" The real answer depends upon many factors but as a starting point for lathes up to 5-inches in centre height with plain bearings aim for a top speed of around 800 rpm - and with roller bearings 1200 rpm. It may well be that higher speeds can be obtained safely, but it would be unwise to go beyond these levels as a starting point. To get a feel for the calculations needed first measure the diameter of the large pulley on the countershaft - say 10 inches. A 2-inch diameter pulley on the motor will give a reduction of 10 divided by 2 = 5 to 1. Divide the motor speed (say 1425 rpm) by 5 and the countershaft will be revolving at 285 rpm. If the lathe has a 3-speed headstock pulley the next higher speed will be twice as fast (570 rpm) and the one below half as fast (142 rpm). This set is obviously a little slow so, increasing the motor pulley to 3-inches in diameter would give speeds of 214, 428 and 856 rpm; that would be a better solution for, combined with the average 6:1 reduction backgear, it would produce a bottom speed of 36 rpm, an ideal rate for the inexperienced to use for screwcutting. If your countershaft pulley is a different diameter, simply substitute the appropriate measurements into the "equation" and experiment with different motor pulley sizes until you have as close a fit to the ideal as you can..

Typical South Bend countershaft unit as used on the 9-inch "Workshop" lathe.
This employed an unusual but effective trick: the motor pulley was a V but the large countershaft pulley was flat.
A V belt was used for the drive - this had plenty of grip on the small motor pulley and, because it was so well wrapped round it, plenty on the flat pulley as well.
The neat, built-on 16-speed countershaft unit of an Atlas lathe.
An earlier form of Atlas countershaft which produced a "deep" speed range - very slow to very fast - without the use of backgear.
Another form of very compact countershaft drive - contained within the cabinet stand of a Logan with a V belt drive going vertically upwards to the lathe above.

Early Drive Systems - Line Shafting
Until the 1930s, and in some cases for very much longer, most machine shops had what would today be grandly called an "Integrated Power System". At the heart of the system was a lovingly-cared-for engine, steam or electric, that drove, via a convoluted belt and rope system, a labyrinthine maze of pulleys hanging from bearings attached to girder work inside the roof of the factory; that part of the drive system held in the ceiling was referred to as "line shafting".
Each machine was attached to the shafting by a wide, flat belt, usually between 1 and 6 inches wide with some sort of ancillary-control system that involved the use of "fast-and-loose" pulleys. The latter was a simple but ingenious system that involved the driven belt running first over a "loose" or free pulley and, from that position, being able to be flicked across to a "fast" pulley clamped to the shaft. Finally, another belt and pulley set took the drive down to a machine on the floor below. Methods of moving the belt were numerous and ingenious from a length of broom handle to sophisticated and expensive controls involving foot pedals, wires, links, bell-cranks and toggles.
Once an overhead drive system had been (expensively) installed in a specially-prepared building, the nightmare of maintaining and constantly overhauling the multitude of bearings and hangers, inconveniently and dangerously located ten or twenty feet in the air, could begin. No wonder Works Engineers clocked-off dreaming of a better solution; their salvation eventually came in the form of the small, high-speed electric motor that was able to provide each machine with its own, independent power source. The tricky installation of a drive system could now be delegated to the machine maker and, besides all the other advantages, if you fell out with your landlord it was possible to pull out of your Victorian dungeon and move across the road, or town, to somewhere both more convivial and cheaper. It also meant that, with an appropriate electricity supply, you could arrange your machines to optimise the production requirements of any particular job and quickly rearrange them again when it became necessary. Meanwhile, George, down the road, stuck in his old-fashioned premises, still had to employ labourers with wheelbarrows to shift 200 lb lumps of cast iron from one end of the factory to the other as a job zigzagged haphazardly around the various machine tools.
Another factor, and now a long-forgotten problem, was the question of light; because there was no electricity to illuminate their interiors Victorian factories had huge numbers of tall windows, glass inserts in the roof and, for preference, were always sited and aligned to make the most of available daylight. The original heavy and cumbersome wrought-iron overhead line shafting and belts did an excellent job of blocking light and even the advent of stronger, lighter and thinner steel components in the mid 1800s did not significantly improve matters - thus the advent of individually-powered machines meant that (just as the light bulb came into use and night shifts started) factories became much lighter, safer and more efficient places in which to work