Evaluating a system - what do you listen for?

Hate:

1. bright high end.  If I cant close my eyes and relax, what's the use?
2 sloppy bass
3. bad room (slap echos, sprung wooden floor, etc.)

Want:

1. I want to hear ample micro and macro dynamics.
2. Soundstage - width, depth
3. Complex music doesn't overwhelm the system
4. tonal purity, starting at middle C
5. bass down to 45 Hz with a .707 damping factor
6. if any driver is better than the others, let it be the tweeter

I listen for phase shift distortion. The fewer crossovers, the less chance of that distortion. 
The faster the crossover slope, the more crossover distortion - phase shift.
6 dB/octave (first order) will introduce far less phase shift than 18 dB/octave.

Listen at high sound pressure levels - 85dB or so, and listen to music that you are familiar with. Do you get Listener Fatigue, i.e. an urge to turn it down?

Finally. listen at low volume. If the speakers don't excite enough air at low volumes, they will disappoint.

The best is not to try and focus on any particular frequency or sounds at first. I am a mastering engineer, and the best technique I use is to almost not focus too much initially. It sounds counter intuitive, but this approach allows you to get an immediate first impression on the overall track, especially if it is one you know. It takes a while to get used to doing this, because you need to not think and you might feel you are being distracted. A good way to practice is to put the track on and do something else nearby, or simply look away from your speakers. Once you have done this for about 5 minutes, you can decide to zero in on various frequencies in a focussed manner. As a rule, vocal and piano are the hardest to get right, so I look for this to sound natural and musical, but not "live".

Try not to look for this "live" sound. Most live sound is amplified and through a distorted PA anyway, so doesn't mean anything by way of comparison unless you are comparing to acoustic live music. Only then is the live approach valid. Even then, many acoustic spaces are not great, so live sound is such a broad church. Better to go with your own subjective preference. 

Psychoacoustics make a big difference too. Looking at a beautiful high end system or listing to one in a glistening top studio where you are overwhelmed by the glamour will often make you want to think it sounds better, so clouds judgement. Simply close your eyes to overcome this problem.

Finally make sure you are in a good mood/same mood each time you listen. Nothing affects your judgment more than how you feel.

I would use the words, realism or holographic, as the essence of what I am now hearing. My last four years have been eye opening to me. I am now hearing very precise locational information, such as where all musicians are, in the case of piano solos even knowing where the high notes are in space, a real sense of ambience in the hall, bass reverb. In addition every note from 20 hz to immediate leading edge, and human noises of the musicians. Sibilants with no breakups and sounding real.

Frankly I never expected to hear what I'm hearing. It is a real thrill to hear performances that I could hear as the musicians are now dead.

I must say that this is all very fragile. Some records don't have this information and sometimes my system is not fully ready, such as having to be on for longer after being off for thunderstorms. But when it is right, wow!