An alarmingly vast question... Requiring an alarmingly vast answer.
Deep breath...
Obviously, emotional response is the main thing.
One of the specific things I like about music though, is the power of a single note. I like when one note does a lot of work, when it's given the room, the space, to ring out and resonate. I also like the opposite sometimes. People often ignore the linearity of the form. They ignore the fact that music occurs over time and occupies that time, and just think of a song as being a thing, an object, a single occurrence.
I like music that has feeling. It's hard to really explain exactly what I mean by feeling, but often it's something that sounds organic and rustic. I'm not that into this long-chased idea of "perfection" in production - I prefer a little imperfection. It's the quirks that make something real. I like music about things. If they're about love, I like them to be actually about real love, and not pop music lust. If they're about hope and joy, I like them to be hopeful and joyous in the face of despair, not sappy or sickly-sweet. If they're about an abstract transcendent experience... Actually, I can't really think of any pop music analogue for that.
Oh, and I like tension and relief. Obviously the entire basis of western music theory is based on tension and relief, but I'm talking on a larger structural scale. I like long crescendos, long build-ups to massive break-downs. I love it when a song pulls me one way emotionally, and then another, and then another.
I listen to albums in full most of the time. Even if there's one song I like less than the others, most of the time I don't skip it. For me, the experience as a whole actually benefits from having these ups and downs, padding out the strong moments over the course of the album and creating a greater effect as a whole. On this note too, I also like long-form musical pieces. I have a number of one-song albums, and albums made up mostly of long tracks, and I often find these more enjoyable than shorter tracks - for one thing, they contain more music, and over the linear time it takes to listen to a piece of music, more things can happen and develop. Even better if there's some coherent musical or conceptual presentation to the album that enhances the effect all the more.
My top played track according to
last.fm is the song Zozobra, from Old Man Gloom's "Seminar III: Zozobra". It is the only track on the album, and weighs in at 27:19. And since last.fm only counts what I play on my computer, if you add up all the times I've listened to that one song, it'd be well over a day and a half. And for some reason that song has proven thus far to have unlimited longevity. Plenty of other records I got around about the same time I will seldom listen to. The song speaks to an epic sense of despair and has a wonderful use of tension and relief, building through multiple sections of sparse slow riffs and drones, a deliciously dissonant one-chord riff that goes through one bar of 3/4 and then seven bars of 4/4, and then at the end, an epic crushing pay off. Not to mention a wonderfully abstract concept of human-as-simian, of de-evolving to the primitive man, of the place of humans in the natural world. This is probably the one piece of music in the world that sums up everything I love about music in one neat package. So if anyone gave a crap about seeing where I'm coming from, that'd be the place to start.