What sonic characteristics matter the most?

Tonal balance is of utmost importance to me. If the bass is boomy, the highs accentuated or the mids recessed I can't even begin to listen. Higher mid anomolies are especially bothersome! It seems that when the tonal balance is off I'm constantly distracted by that imbalance and can't fully immerse myself into the music. Psycho-acoustical issues I suppose, a gross deviation from the natural acoustic?
After that it becomes a close call, everything matters and if any specific characteristic begins to annoy I have to fix it.

timbres. It has to sound like the instrument that is being reproduced, in a natural sounding, effortless way. That allows me (At least for me) to suspend disbelief. I can forgive everything else if this is done right. This is why I gravitate towards planars, tubes, and analog. For me, it just gets the "tone" right.

I'll agree with Aroc and add bass done properly which to me means proper weight. Also agree that planars tend to sound more real than cones for accuracy.

This is a great question! It should have received more response. I admit, I don’t know how to explain what I like with the right words but I can try to give you an idea at least.
My first priority is the 3D/holographic imaging/soundstage. When I hear Bon Iver as if his voice is in the air right in front of me, I am happy. However, my system also needs to give me the soul of his voice. I am not after perfectly clear and detailed sound.
Separation of the instruments is another thing I value. I like to hear the piano way to the left and violin all the way to the right.
Romantic/natural sound is also another one I go after. I want to be able to relax when I hear the music. Bright sound, I cannot do.
Bass, I am not too crazy about. My speakers don’t have to go all low low. I can live with limited bass.
If I have to tell you few brands maybe you may have better idea about my taste.
Magnepan, Harbeth, Sonus Faber, Pass Labs and you will be surprised but amps with Tripath chip amps. I am not too familiar with their current models but these brands always gave me good sound. Whenever I owned these brands at my home, I stopped criticizing my system and just listen to music.
Currently, I have Harbeth SHL5 (not Plus), Topping TP60 Tripath integrated (I know you are like this guy doesn’t know anything:) ), Museatex Bitstream DAC. I will replace the DAC and just got LTA MZ2 pre and will DIY board amp with double Tripath TA2022 chip. Yes, I like Tripath chip that much:)
If you want to impress me with your super duper high-end brands which sound bright and detailed, no thanks!

For me, the most important aspect of a system is in its ability to reproduce extremely fine Micro Details
e.g. those details that convey…
- the extremely fine venue acoustics and reverberations
- the subtle timbres of an acoustic instrument
- the subtle harmonic tones or the “grit’ in an artist’s voice
- the controlled breathing of a woodwind artist as they play
- precise placement and complete isolation of the artist within a complex image, such as an orchestra
- an image that is larger than the listening room and which envelopes the listener

That’s what I listen for. I’ve stopped listening for the "conventional metrics".

Without micro details, a recording is flat, lacks vibrance and realism.

The strange thing is - my components didn’t need upgrading at all - I just needed the right cables :-)

A close friend once asked me "how many speakers do you have connected to the system now"
- I had to show him the left and right speaker cables to convince him there were only two speakers operating

Regards - Steve

Coherency, and the "system" seeming to not be there when the music is playing, as otherwise when it's not playing I'd trip over things. 

That’s what I listen for. I’ve stopped listening for the "conventional metrics".   Without micro details, a recording is flat, lacks vibrance and realism.  The strange thing is - my components didn’t need upgrading at all - I just needed the right cables :-)

@williewonka  +1