Say your 150mm engineer's pocket ruler has markings at 1mm intervals so you can read to the nearest 1mm.
You have available measurement values of 0, 1, 2, 3.......148,149,150mm

We say you can 'resolve' the distance measured to 1mm.

So say you measure 127mm.

Now this distance is 157mm on a map at 1:2500 scale so represents 317.5 metres on the ground.

The difference you can resolve on the ground with your ruler is 1mm x 2500 ie 2.5 metres.

So you apply your 8 bit ADC which has 256 values available to some voltage V which means you scale the voltage into lengths of V/256.

So if you were measuring 10 volts your minimum resolution would be 10/256 volts in voltage measure or 1 bit in bit measure.

If you say the resolution of the signal is 256, then the unit is "dimensionless".

If you say the resolution is 8 bits, then the unit used is "bit", but is signifies 256 levels that are "dimensionless".

If you say the resolution is determined by a least-significant-bit voltage of 20 mV, then the unit is volt, although that's not really the unit of the resolution.
Volt here is the unit of the least-significant-bit voltage.

If you say the resolution is 20 mV, then that certainly does not mean that you have 20 levels.
It would be shorthand for saying that the least-significant-bit voltage is 20 mV.

Engineers are usually a bit sloppy in how they write things down.
However, the key is that you understand what was intended.

The key certainly is that you (or whoever reads what you write) understands what is meant.

To a physicist, the only meaningful resolution would be 20 mV.

When you refer to 8 bits or 256 levels, a physicist would consider that the dynamic range (the resolution always being 1 bit (the least significant), or 1 level.

In case of doubt give a complete description, specifying number of bits (or dynamic range in decades) and the LSB resolution in mV (or nA or whatever other analog input you use).

Say you had a signal from a microphone, digitised to 8 bits and passed it through a pre-amplifier then to an amplifier and then to a loudspeaker.

Your signal now would never have greater resolution than 8 bits at any stage in the chain, however the voltage resolution would alter dramatically at each stage.

Hmm, I just looked up what the word "resolution" actually means and what its unit should be.

It has a lot of meanings, but as I understand it it's not a quantity that you measure, but it's a concept.
The relevant dictionary meaning seems to be:

Typical usage in a sentence is that the resolution is "described" or "indicated" by something.
So the resolution of a signal is "indicated" by the number of levels that can be distinguished, or by the number of bits.
The resolution of a monitor is "described" by the number of horizontal and vertical pixels.

Here's a couple of usages on wikipedia (bolding mine):

Hmm, the digitizer that converts the microphone voltage into an 8-bit digital signal has a defined resolution of x mV. I think so far we agree.

Then you are skipping a step, as the digital signal has to be converted back into an analog voltage that then passes pre and power amp.

What happens then is a bit more complicated. Assume that both amps are absolutely perfect. Then the signal quality will be conserved (it can't get better, no matter how expensive you amps are), and in the end you still have an audio signal with the dynamic range of the original 8-Bit encoding. The actual voltage levels will vary of course.

Now assume that you have a really crappy amp with loads of noise. For the sake of argument, let the DAC put out 8 bits with 20 mV smallest steps, i.e 256*20mV = 5.12V max.
Now let the preamp have a noise level of 100mV (for sake of argument, not even physicists built that bad amps). The resolution (in mV) cannot be better than that. Then the dynamic range will be 5.12V/100 mV = 51 which is closer to 6 bits.

Note that audio resolution is not included in the section for measurement resolution. For digital audio, my guess is that there is a convention for the maximum voltage, so that the resolution ends up as this reference value divided by 2^(number of bits). For analog audio, Dezibels (dB) are used, which define a dynamic range relative to a 0 dB reference signal.

So for two monitors, one 5 times as big as the other, but with the same pixel resolutionthe pixels of one will be five times bigger than those of the other.

For a physical instrument, the resolution is defined as the smallest separation of two signals it can distinguish=resolve. As such, the resolution is an absolute value in mV, nA, micrometers, millidegrees, millikelvins, or whatever units you happen to be measuring.

The dynamic range is the ratio of the largest to the smallest signal that can be measured.

If the largest signal is fixed, then the resolution can be calculated from the dynamic range. If the absolute value is pretty much meaningless because the signal is scaled back and forth (as in audio), then only the dynamic range is of interest.

These definitions are commonly used in the world of physics. It appears that in electronics (and in particular audio) these definitions are used more freely.

I would call that pixel count, not (pixel) resolution, but you are correct that in common use it *is* called resolution (what I call resolution is listed as dot pitch or pixel density). This makes sense if you change the settings on your video card while using the same monitor. Displaying a different number of pixels on the same area then changes the resolution.

For printers, on the other hand, the resolution is defined in dpi (dots per inch). If you double the size of the print, keeping the same resolution, then you quadruple the number of pixels.

I guess the message to take home is to be as precise as possible in your writing.

BTW, the current that goes through the loudspeakers is still analog. By remaining digital as long as possible you avoid signal degradation and loss of dynamic range. But you know that :-)

Bear in mind that for any A/D converter the step size is compared against an internal reference so is fixed.
In particular it does not depend upon the input voltage.
Each step will be in exactly the same place (at the same voltage) regardless of the input voltage.

So saying that the max input voltage or full scale input voltage equals 2000 mV and there are 1024 (8 bit) steps or that the resolution is 2000/1024 mV is one and the same thing.

Can't help responding, nitpicker that I am.
Sorry.

1024 is 10 bits (not 8 bits).

If there are 1024 levels, then there are 1023 steps.

It still depends on the specification of the chip what the stepsize is.
It could both be 2000/1024 mV or 2000/1023 mV (or even something else).
I, for one, certainly wouldn't make an assumption which one it is.
Only when I see it on an oscilloscope (or something like that) will I believe which one it is (for a specific chip).

However there is only one possible interpretation of the process that takes an input voltage and replaces it with a unique digital code with one of 1024 possible values.