What exactly is critical listening? Who does it?

There are different types of "critical listening" for different purposes. In audiophile land, I think it is often meant to describe listening for shortcomings in the presentation of a system or given component. But it can also involve listening to compare different pressings of the same record, to evaluate their respective merits. In musicology, there are also different types of critical listening- one might be to analyze motifs or how it fits into the overall body of that composer's work, its influences, historical context, etc. Forensic musicology often involves picking out allegedly infringing phrases or elements and finding historical precedent for their common usage as building blocks by other authors. I think you need to provide context to afford more meaning to the term. To me, I most often associate it in home audio reproduction with listening to where a system or component reveals its shortcomings. 

Correct.  Get busy now and let us know your findings. Chop chop!  Enjoyment is right out! 😉

Myself:  guilty as charged.  Call me Mr. Critical.  Just be nice about it.  Not too critical, please.  

Correct, critical doesn't necessarily involve judgement. I mentioned in another thread how in recent years more focus on qualities of the recording itself. Producers/engineers play a major role in bringing us these works of arts, I've learned to turn off the judgment cap, just let the recording come to me, much more satisfying listening sessions with this. You still the warts, they become curiosities rather than irritants. 

"critical doesn't necessarily involve judgement."

Correct:

Critical

I define critical listening as focusing only on the music-maybe not the precisely correct meaning. Lots of people-my daughters for example-always have music on, but its in the background, while they are doing other things. I bought them both a Morel Biggie to replace Alexa and they consider it high end.

When I listen critically, I try to devote at least an hour and try to clear my mind (with my mind, not a lot of deep thoughts to get out of the way). I'll admit that I enjoy the music, but also enjoy judging how its reproduced both in terms of the engineering, and by my equipment. For me, the performance of the gear is an integral part of the experience. I realize I could be voted off Audiophile Island for that damning admission.