What exactly is critical listening? Who does it?

I thankfully don't engage in audiophile nonsense such as "critical" listening. that is such BS it is funny. As if we were in the studio when the recording was taking place. the self fellation that occurs in these circles is comical

Suppose I want to compare the sound of the same radio program via streamer/DAC vs. broadcast (FM tuner). I simply switch between input X and input Y on the preamp. That's an example of critical listening, and it's neither nonsense nor self-flattery. Or I want to compare the sound of preamp A vs. preamp B into the same amp (one on balanced, the other on RCA). Just switch between the two inputs by pushing button on front panel of amp. That's another example of critical listening, and a useful one. The list of potential listening tests is pretty much limitless, but of course, once satisfied, we lean back and enjoy until some other aspect of the system warrants our critical attention.

To me critical listening is focusing on things like sound staging, pinpoint image location, and all the other esoteric things that a typical audiophile cares about.  Casual listening is not focusing on that but more so just enjoying the music.

If you are focused solely on listening you are listening critically. The more distraction the less critical the listening.  

Critical listening is done in order to attempt to get the sound of your system to sound as good as possible to your ears. Once you're done with the room soundproofing and acoustic treatments, then come the equipment upgrades and only then you finalize it with subtle sound changes with various cables, fuses and isolation products. Each one of those changes will easily be noticeable and that's when critical listening is absolutely necessary. The hope is that eventually you never need any more critical listening. Audio nirvana. Am I there? No. Am I close? Definitely. It's a journey, just like the spiritual one.