>>Visually DPI represents "dots per inch" - it's a measure of "density" in
printing. The higher the DPI the better the quality of printing - right?

Exact. It's a density. The definition of a "resolution" is "the size of 
the smallest element the instrument can measure,
-- if the instrument sees or measures things -- or generate, if the 
instrument creates things.
It is closely related to the density, of course, but it is not formerly 
the same thing.
Thus, the unit of resolution for instruments like printers, monitors or 
scanners would be in inches,
or 1/1000 In. For instance, a printer with a density of 1000 DPI has a 
density of 1/1000 In.
Note that on some optical instruments like cameras, or human eye, the 
resolution is measured
in angle units. The human eye has a resolution of about 1 minute angle.

 >>On a monitor the "pixel" is in one way a measurement of size - as in "how
many pixels can I get on the screen".

Ok, now "how many pixels", this is called the "definition". This is the 
maximum number of different elements
the instrument can measure or genrate (ie: lines in TV transmission)
Of course, the more pixels you have to represent an image, the better 
the quality will be.
Note that:
density = definition / size of the working area
resolution = size of the working area / definition
then density = 1 / resolution

 >>When I look at an image in Fireworks I
get a measurement of pixels at the bottom - 800 by 600 for example - 96
"pixels per inch". That's a size measurement.

For the pixels, yes, but one should never should forget that it is a 
virtual image.
Normally, a virtual object has no size.
Now, if you make a physical representation of your virtual image, it 
becomes a real image,
and its size will be nb.pixels * resolution of the printer,
or nb.pixels / density of the printer.
Talking of pixels per inch in a virtual image just does not make sense.
Where are the inches we are talking about ?

 >>This setting (as I understand it) is really
related to the lossy compression that is used and dictates how aggressively
the file is to be compressed.

Exact, and it has nothing to do with pixels, or density.

 >>But this is where non-imagy folks like me get confused. I DPI is not the
same as "resolution" what is the proper term?

In my book it is "density".

 >> Is it "bit depth" or the number of colors used or what.

Yes, it is the numbers of bits used to express color, thus the magnitude 
of the numbers that can be used
to express the color. Then one should finally call it the "color 
definition".

 >>How do you reference the quality of the image
on the screen without using the word "resolution"? And if resolution IS the
proper term for the "quality" of an image, then doesn't it have at least
some correlation to DPI?

You cannot express the quality in DPI or any other kind of unit.
The quality depends on many factors like the pixel definition, the color 
definition, the compression level
the precision and fidelity of the display or printing equipment, and 
finally, the quality of your own eye !
The overall quality is a chain of elements, DPI is only one of them.

 >>I need education on this point. What is the proper way of referring 
to the
"quality" of a digital image?

There is no really a unit to measure "quality", one only can take in 
account the quality of each
element in the chain.
Just like in a old analogic sound equipment, the overall quality depends 
on many element, from
the musical instrument to the auditor's ear, including the recording 
system, the vinyl record pressing,
the piezzo pick-up, the speaker system, etc.


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