The transformer.
Makes the 24 volt control power everything else runs on. When it dies, the thermostat goes dark.
Line voltage runs the motors. Everything that controls them, the thermostat, the contactor coil, the safety switches, runs on 24 volts.
The transformer makes that 24 volts. It is usually mounted in the air handler or furnace cabinet, and in most houses it is the single point of failure for the entire control system.
When it dies
Everything on the control side stops at once. The thermostat screen goes dark if it is hardwired, nothing responds, and the system is completely inert while the breaker is on and line voltage is present at the unit.
That is why a blank thermostat is not automatically a thermostat problem, and why the dead system guide puts the control circuit on the list.
It usually did not fail on its own
This is the part worth understanding, because replacing just the transformer often means replacing it twice.
Transformers mostly blow because something downstream shorted. A thermostat wire chafed through where it passes a sheet metal edge, a rodent chewed the cable in the attic, water got into a junction, or a control board failed shorted.
Fit a new transformer without finding that, and the new one goes the same way, sometimes immediately.
Many systems have a small fuse on the board protecting the 24 volt circuit, which blows first and saves the transformer. A system that keeps blowing that fuse is telling you the same thing.
What a proper diagnosis looks like
Confirm line voltage in, confirm 24 volts out, and if there is none, find the short before fitting anything. That means checking the thermostat wiring run, the safety switches, and the board.
It is not a long job, and skipping it is how the same repair happens twice in a fortnight.
After a power event
Surges on restoration take out transformers and board fuses fairly readily, which is why this comes up after monsoon outages. The power outage guide covers what else the restoration can damage.