OK. My attempt at optical comparator instruction:
I hope this makes sense and helps a little.
Basic O.C. Operation 101
The obvious feature of an optical comparator is that a magnified image is projected onto a screen, like a microfilm projector at the library. The image can either be a silouette or a reflected image, depending on which lighting mode you choose. The projection screen can be interchanged, or a transparent overlay placed against it, to use different measuring aids for the feature you are checking. The most common and versatile projection screen is a simple cross-hair.
You place the "widget" on the table in front of the lens, focus the image as crisp as possible (as simple as that is, you'd be surprised how many folks I've worked with don't know how to focus an image).
Then you move the table until the cross-hairs on the projection screen are dead-on your first point of reference on the widget. At this point you zero your dials or DRO. Without a DRO this is a little trickier. Since the method varies from machine to machine, I leave that to you to figure it out.
Then you turn the cranks and move the table so the cross-hairs are dead-on the next point to find the distance between this second point and your first reference point. This method of measurement does not depend on the accuracy of the lenses/magnification since you are looking at the exact same spot on the screen (see note below). The table moves and you read the micrometer dials to find how far you moved it.
End of lesson one.
Bonus factoids:
Overlay Measurement Method
Most comparators are equipped with very well-made optics so the magnification is very accurate. But you still should check the comparator's magnification against some kind of precision standard before you depend on overlays/projection screens to tell the truth. An overlay that is printed perfectly to 50X scale will not quite groove with your projected image if the comparator's magnification is 50.03X and may cause problems for you. The solution is to match the scale of the overlay with the actual magnification rather than what the lens says it is supposed to be. There are other factors with optics that can cause distortions on different areas of the projection as well.
Optical comparators have a chain of elements that need to be aligned/calibrated for them to perform their best. I've never looked into it, but apparently it's a tricky thing to do it well. There are engineers that make a business out of calibrating O.C.'s for a list of clients around town, and they aren't a cheap date. My guess is that for home shop use a satifactory job could be done once you learn the basics. I hope this is true anyway, since I have one in my basement that could use some TLC. Hopefully it's not much tougher than collimating a telescope. Maybe Evan has some experience with this.