Before the trans-axle was invented, the transmission, differential and driving axles were all separate assemblies. All these separate components do a different job (more on that later) and were as you can imagine quite bulky, getting them to fit into a small area was a challenge.
The trans-axle is compact and combines all the functions of the separate assemblies, it's main function is to transfer engine power into forward motion. Depending on how sophisticated it is, you may have reverse motion and variable speed control.
Trans-axles are the normal setup in all front wheel drive cars, and most people will refer to a trans-axle as the transmission, or transmission assembly.
All lawnmowers will have axles but unless they're power driven they won't have a trans-axle. Most walk behind self drive mowers are rear wheel driven and have the trans-axle mounted at the rear, but some craftsman mowers are front wheel drive and so the trans-axle is mounted at the front.
Rear drive is better for hilly ground where you need good traction, and front drive is better for a yard with lots of shrubs and flowerbeds, they're more manoeuvrable.
Higher end self-drive walk-behind mowers will have variable speed control.
Lawn tractors will have the trans-axle at the rear of the mower, it's usually pretty bulky and well defined, with a fat wheel at either side.
Lawn tractor trans-axles are either manual or automatic just like a car. The gear driven trans-axles have a clutch pedal and a gear selector, the brakes are also mounted on the trans-axle.
The auto version has a variable speed forward and reverse pedal and is usually a hydro-static (oil pressure driven) or CVT type.
The highly manoeuvrable Zero turn mowers are a little unique in that they have two hydro-static trans-axles, one for each rear wheel. This set up allows them turn tightly, just like a shopping trolley.
Trans-axles are driven as you know by engine power. A belt from the engine pulley, mounted just above the blade, transfers power to the trans-axle pulley.
Turning the trans-axle pulley on some trans-axle types causes the mower to move, on other types, a lever will need to be operated also.