What are the functions of gear train on a lathe machine?

All gearboxes work the same way: there is a gear on the motor, a gear on the thing you're trying to drive, and sometimes a set of gears between the two. The ratio between the gears, plus the RPM of the motor, determines the speed of the workpiece. If the gear on the motor is larger than the gear on the workpiece the RPM of the workpiece will be higher than the speed of the motor. If the workpiece's gear is larger, the workpiece will turn slower than the motor. Given that, though, I've never seen a gearbox on a wood lathe. Wood lathes use belt drive for two reasons, the most important being safety--if you get a catch, a workpiece on a belt driven lathe won't damage you as badly as a workpiece on a gear-driven lathe. Also, belt drive won't feed vibration from the workpiece back into the motor and damage the bearings. Gear drive is used on metal lathes.