Q:

What is the "propane test" and how is it carried out?

This test is often mentioned in the context of figuring out what's wrong with a car engine's operation (misfires, hesitation, poor fuel economy, etc.)
What is this test and what does one need to perform it?

A:

The propane test is used to find vacuum/air leaks in the intake of an ICE
I used to use this test until I lost my eyebrows.
You use propane to find vacuum leaks by applying the propane to suspected air leak areas of the intake of a combustion engine.  You are looking for air leaks between the carburetor or throttle body and the cylinder head.  An air leak before the throttle body is not an air/vacuum leak because there is no vacuum that will effect mixture before the throttle body or carburetor.
Here is what you need.

Small Propane Tank
A metering mechanism designed to screw onto the propane tank
4 feet of rubber hose
Maybe, some duct tape

Take your hose and slip it over the end of the torch.  If it doesn't fit snuggly, use your duct tape.
You will disconnect your TPS (Throttle Position Sensor) before you start so the ECU goes into an closed loop mode and isn't trying to constantly and minutely adjust the golden ration 14.7:1 AFR while you are trying to get this done.
While the car is running turn the propane on and start moving the the end of the rubber hose to areas where there is suspected air leak.  You will want to linger over suspect points for a bit to allow any oxygen to get moved out of the way.  When the engine starts to suck the propane the idle will smooth out, misfires may even go away.  This works because the vacuum leak is sucking in propane and not oxygen.
It's pretty effective unless you do something that becomes dangerous. I had another mechanic leave a propane tank on and come to bay to ask for help. He was working on his personal car. He left the propane on, the car was off. He wanted to reset his ECU and disconnected his battery, it sparked......you know the rest of the story. I never used the technique again after that. I think stupidity drove the incident but nonetheless I've only used the smoke trick and the the WD-40/Carb Cleaner trick ever since.
I'm not going to tell you what this picture is for

A:

A version of the propane test can be used to test for bad cats
This is for vehicles equipped with OBD-II readers and both pre-cat and post-cat lambda sensors

Items needed

propane supply
OBD-II reader

Safety Warning : DO NOT IGNITE THE PROPANE SUPPLY
The propane is there to just induce a rich condition. More on that to follow.

How the test works
The test is designed to test a catalytic converter's ability to "hold" oxygen for a short period of time. The inability of the catalytic converter to store oxygen for at least a few seconds is an indication of a cat that is not functioning as it should.

Steps

Get the cats up to operating temperature.
Monitor lambda sensor voltages with the OBD-II reader.

Pre-cat sensor(s) should cycle between lean and rich
voltages (0.1-0.9 V on narrowbands)
Post-cat sensor(s) should hold rich (0.7-0.9 V on narrowband)
.

If the post-cat sensor(s) cannot hold rich, this is indicative of a bad cat - no point in proceeding further.
If the post-cat(s) hold rich, continue to Step 2.
Add propane to the intake.
The pre-cat lambda sensor(s) read rich after a few moments (ditto for post(s))
Stop feeding propane after a few moments of steady rich pre-cat voltage(s).
This is the important part of the test:

The pre-cat(s) should read lean after a few moments (because the short-term fuel trim hasn't had the chance to do its thing)
If the post-cat(s) read lean almost immediately (< 2 seconds), the cat is bad
If the post-cat(s) read lean after a few seconds (e.g. 6 seconds), the cat is healthy 

