Variable Autotransformer, also known as VARIAC
Tim Reese posted a great article about it on a Harvard site.
I bought mine on Facebook marketplace from a retired engineer for $60 CAD and experimented with it. Sometimes I think it is a shame that I'm only learning what these devices really do decade after my degree.
What does a variac do?
You can use a variac for testing equipments by slowly adjusting the voltage.
Variacs provide a voltage-adjustable source of alternating current (AC) electricity. Usually plugged into a wall outlet (120 volts AC at 60 Hertz in the USA), they have a knob-controlled output that can range from 0 volts AC (VAC) to about 140VAC, depending on how the winding is connected (see below). Variacs do not provide isolation from the wall outlet (the line voltage) as does a true transformer; thus are not suitable for directly powering low-voltage devices. They are typically used in tube audio for powering experimental circuits and for equipment testing and repair, not as a permanent source of voltage-shifted power.
Variacs are not isolated
There is only one winding on variacs compared to a transformer. Transformer is the device that you plug into the wall socket to power your low voltage devices (i.e. phones, laptops etc.). Transformers are isolated and therefore protect the low voltage devices. But, a variac is not isolated.
What order to follow?
Isolation transformer first and then the variac. Putting a variac in front of the isolation transformer can cause poor voltage control, especially at lower voltages.
For what it's worth, I've always wired the Variac to the secondary of the isolation transformer. Been doing it that way since the 1960s.