What exactly is critical listening? Who does it?

It is at first impression a matter of semantics. Critical can obviously mean at least two things - one connotes negativity - ie a person being critical points out negative facts, [their] observations, or perceptions from a real or supposed (usually subjective) ideal.

The second meaning/use is a variance from an objective standard. Ie. a business plan or financial investment analysis requires a critical review of all known variables, and some accounting/adjustment for unknown future impacts. All engineering and design requires critical review for many reasons.

Socrates was attributed with saying "an unexamined life is not worth living". Meaning (I think) that all people should consider how to examine and improve themselves (physically, emotionally, educationally, financially, spiritually, etc.) - to at least be working a little over time toward becoming a ’better’ version of themselves in their roles as father, employee, neighbor, mentor, professional, etc.

I don’t think of critical listening as having any ’negative’ connotations. I will choose [when] to notice detail, soundstage, tone, emotion of the singer or music, fidelity, warmth, yada yada. I can listen to music in the car or home completely as background music, but I can also listen ’critically’ - and remember how the song ’felt’ when I heard it live in a concert as a teenager, and how it sounds now on a personally curated hifi system.  All good. It becomes ’negative critical listening’ when a) you can’t appreciate any level of the playback for what it is, and create some ’comparison’ in your mind when you hear playback that is cognitively dissonant from what you think it should be, or usually worse, a sound or experience that is out of financial reach. Comparison or 'criticality' in that sense chases that mythical dragon of some satisfaction that is unachievable - in the comparison mindset. Like Einstein said "we cannot solve the problems with find ourselves in, with the same type of thinking that created them". 

I see a couple of situations where I use critical listening - extensively when purchasing new components and again during set up in my home listening area. 

For me the most important thing is to have a set of 6 or so musical tracks that I know very well, and that represent the different parts of the audio spectrum. Bring with or have AD stream, so that you can listen for reproducible instruments/sounds from the component being auditioned. Same thing when setting up a listening area (moving things around, adding a new component). 
 

Critical listening for me evolved into musical enjoyment once things are just right. 
 

At one time I had this romantic system, totally drawn to the music, and then at some point I became aware I was only listening to certain types or recordings. This probably went on for a year or so prior to this awareness. So I started listening to a wider range of recordings, the defects and flaws that had been there all along suddenly became salient and I could never NOT hear those flaws from that point on. Point is I had subconsciously stopped listening critically, had a wonderful time while it lasted. 

So that system goes bye bye, build the next system only to satisfy my critical listening mode. This goes on for some time until I come to realize this system not emotionally engaging, this system then goes bye bye.

So now I've been to both ends of the scale, ever since that realization I've finally been able to build a system that engages me from the critical, analytical and emotional states or perspectives. When you finally experience engagement or appreciation for all three states of consciousness you find contentment. I purposely seek engagement on all three levels, never want to go back to the need to ignore some listening perspective or avoid conscious, aka critical listening. 

If a system is utterly musical and pleasing to my ears, I don’t think how it sounds. We keep thinking about its sound (critical listening) because the system is not musical and unpleasant to our ears and it became an object.

Can you think about how it sound when you listen below music which, I think, is more musical than the original music?  Alex/Wavetouch audio

Eric Clapton - Tears in heaven

Original music 

I agree with @ozzy62’s comment concerning “critical listening” could mean something different to each individual.  For me, critical listening is when I am simply listening to minute details in the music to see if a new product I am auditioning is improving the sound quality.  I’m critical listening so I can make a decision on a product I’m contemplating on buying.  I imagine most of us would do that before we spend money on our gear, if we have an opportunity to audition a product.  So, 99% of the time, I’m NOT critical listening, but enjoying the music because I finally got my system dialed in and I have no desire to audition or replace anything.  That’s what critical listening means to me,