Q:

What is one "stop"?

I always hear this term, e.g., 

I had to go down one stop
Increasing X by Y raises Z by one stop
I turned down the flash/the light two stops
This lens/sensor/strobe/Photoshop tweak raises X by around one stop

That one too!: "Around one stop"... Is there a 0.85 or a 1.13 of a stop, to be exact?
Is this (always) the same thing as an f-stop?  I am so confused!

A:

A stop will halve or double the amount of light, depending on the direction and that could mean the amount of light reaching the sensor or how sensitive the sensor is made to the light that is reaching it. 
So, for example, to reduce something a stop, I could go from ISO 800 to ISO 400, or I could go from a 1/500 to 1/1000 shutter speed, or I could change the aperture from f/2.8 to f/4. Going opposite direction on any of these would increase the light by a stop.
An f-stop is the term used to describe the aperture positions on a lens. It's the basis for the more general term of a "stop" when describing the amount of light for the exposure.

A:

An alteration of 1 stop either halves or doubles the exposure of a photograph. And you're right, it's exactly the same thing as an f-stop.
Imagine you've pointed your camera at a scene and it's recommending the following settings:

A shutter speed of 1/100s
An aperture of f/5.6
A sensitivity setting of ISO 400

We could make the photo appear twice as bright by doing any of the following:

Doubling the shutter speed to 1/50s
Doubling the area of the aperture by increasing it one full f-stop, to f/4
Doubling the sensitivity setting to ISO 800

Any of these could be described as increasing the exposure by 1 stop.
Likewise, to make the scene appear half as bright, we can reduce the exposure by 1 stop by taking one of the opposite steps

Halving the shutter speed to 1/200s
Halving the aperture area by reducing it one full f-stop, to f/8
Halving the sensitivity to ISO 200.

Note that we use the word "stop" even when we're not adjusting the f-stop (aperture) setting. It's just become a general term to mean a factor of 2.

A:

It looks like you already have quite good answers, except for the
possibility of fractional stops. The answer to this last point is yes,
fractional stops do make sense: 0.85 stops varies the exposure by a
factor 1.8, and 1.13 stops is a factor 2.2. More generally, if you vary
the exposure by a factor k, then the number of stops is
log(k)/log(2). In other words, k=2^(number_of_stops).
As an example, below is a table to convert an exposure variation (given
as a factor, e.g. “× 2.5” means 2.5 times more exposure) to a number of
stops. It goes from -2 to +2 by third-stops.
exposure    stops
-----------------
× 4         +2
× 3.2       +1⅔
× 2.5       +1⅓
× 2         +1
× 1.6       +2/3
× 1.25      +1/3
× 1         0
× 0.8       -1/3
× 0.64      -2/3
× 0.5       -1
× 0.4       -1⅓
× 0.32      -1⅔
× 0.25      -2

