Measuring Print Quality
Sheet fed Offset and Gravure are high-quality printing presses that are used to achieve the highest precision registered images.
The quality of a printed image cannot be easily judged. The precise technology that is used for colour is not reproducible. This might seem strange; but, it's true. No two manufacturers of colour quality measuring equipment can reproduce another's findings. No two instruments agree. It's troubling.
The best attempts result in some close readings.
Printer's (the people) use test targets which are also called printer's marks. There are various ones in addition to finishing (trim, fold, score, punch, emboss, etc.) for different purposes.
A geneva star is a pattern that can be visually examined to determine resolution, press sheet movement during the impression, direction of the unwanted movement (different controls for different directions) and a couple of other esoteric things like double-impression and slur (twist).

When you see one, look to see if there is a small clear spot in the centre. No, then there was a printing problem. There's a specific pattern for each problem diagnosis.
EDIT: Here's what to look for, close-up.

This must be repeated for each plate.
Colour bars are series of small ink patches of 25%, 50%, 75%, and 100% (solid) for each plate which are read by machine to tell the variation from ideal. Dot gain (change in contrast) is one of their purposes.
Register marks indicate how well the various plate images fall on top of each other.
Misalignment of the printing plates is one of the basic problems with colour. As a matter of practicality, If you are ever asked to approve a press proof, the first thing you should check is that the alignment (registration) is PERFECT. Nothing can be judged until that detail is nailed-down. Any time you hear, "We'll get the registration. Just approve the colour." Pack up and find another shop. They don't know what they're doing and have just told you so in as many words.
There are other targets developed for specialized troubleshooting of various press problems.
We rely on technology today for mechanical press control. The early machines were ink densitometers, now, spectrophotometers are used that are more accurate. There are various alignment and calibration procedures that have been developed to ensure the reliability of the devices used.
Here are some other ways that we control printing quality "By the numbers."Printing Industries of America
EDIT: I forgot to mention that since printers have no control over what they're asked to print, they have adopted indirect methods to assure that whatever you want to have reproduced will be done faithfully even if it means that if you give them garbage, it will be reproduced exactly as you gave it to them.
"All mistakes will be faithfully reproduced." hung over my desk for years.
This is why you'll find various printer marks put into the live area of the printing plate but outside of the trim (discarded part) of the flat (the printed piece before being finished).
These test target originals are perfect images that will degrade when reproduced. The amount and kind of degradation indicates the corrective action that must be taken.
What does all this mean?
While I've been writing about how press operators judge print quality, you can use the same techniques they use. Learn how to use the printer's marks. Learn how to do a press check and what to look for in proofs, colour management, get yourself a loupe, or borrow one to look closely at registration. Check to see how closely the proof matches the press proof/run.