What is "Hotel Mode" on a turboprop engine?
Hotel mode is a feature on ATR 42/72 turboprops replacing an APU by locking the right-hand #2 engine (PW100) with a 'propeller brake' while allowing the turbine, and therefore also the generator, to run, providing electrical power and bleed air.
ATR describes this as the following:

The propeller brake is fitted on a countershaft on the RH engine reduction gearbox in order to stop the propeller (and the power turbine)

HP (High Pressure) spool drives the DC generator.

Bleed pressure is available downstream the HP compressor and supplies both packs.



This picture should give you sort-of an idea of how it works:

If I gather correctly, the propeller drive shaft is locked, and the (HP) compressor shaft is free to run.
Here's another pic of a PW100 engine:

Hotel Mode seems to be generally disliked for a number of reasons:

It's very loud, especially if you're lifting baggage next to it
It consumes a lot of fuel, since you're running a full engine.
It could be dangerous if the propeller brake is in poor condition and suddenly loses its grip.

This configuration is rare, if not unique. The rival Dash-8 has a separate APU, as does the Saab 2000. It was an optional feature on the Fokker 50. The smaller Jetstream 41 only has a battery.
Source: SmartCockpit ATR-72 documents.