Hmm.. so be it, I have not much time now but ok.
Autotransformer looks like any other transformer for a first look. It's smaller than a transforme of the same rated power. Have no schematic I am allowed to post, and have no time to search for one in net so I'l try to explain it.
Typical power autotransformer works in grid as unit connecting high voltage networks (110/230 kV in example). The closer are the primary and secondary voltages the more you gain compared to a trasnformer of same rated power.
In general, it is construced just like a transformer (core, windings, insulation system, cooling, tank, tap changer and others). Main difference is how the windings are connected.
1. You have tetriary winding, it is wounded mainly as the one closest to the core. Connected as delta winding. It supports the main flux in core with higher harmonics, so the flux in core is much more like sinusoidal. It is not indicated on schematic I've uploaded.
2. Second from the core is the low voltage winding (secondary). It is the lower one (where I=I2-I1 current flows) on picture (a) in one of my previous posts.
3. Third one is the high voltage winding (current I1 flows thru it).
4. The last one is the regulation winding. Regulation winding is connected between end of primary winding and point A1 (it's only one of the possibilities, it can be placed elsewhere. For more details, please look in books). It has coils which can be connected in many configurations by device called tap changer (for more details
http://www.reinhausen.com). It can be done under full load. Now, imagine You have 100 turns in primary winding and 20 turns in regulation winding. What are the possibilities of regulation ? For some types of tap changers You can have 100-20 to 100+20 turns (plus all sort of steps between them) because You treat primary and regulation winding connected in series as one winding (Imagine solenoid. It has 120 turns, all connected in series so the current value is the same in all turns. 20 turns can be connected as opposite or in the same direction to this 100). You change the turn ratio, You change the voltage output. I hope that is clear.
The main prinicple of AT working is the same as in transformer, no black magic here. You may ask, how it is that currents I+I1= I2 and flow to the output. I'd say this is done by proper connection of winding end terminals (another long story here).
Sorry, but this is complicated stuff. Discussing or explaining it in details will take long time which I don't have. That's why I'm sending You to books, when You'll rethink this and have some more detailed quetions we can talk.
EDIT: from profile of Your questions I see that You're not interested in power auto transformers. Note that designs vary, small 'lab' AT's are made different than big power grid ones.