Diamond all the way

I will join the camp about the vocals usually making or breaking a song for me. The music would have to be pretty spectacular for me to get past vocals that I dislike. I'll never forget the first time lsitening to Mortification when my brother bought it on a whim (not knowing it had all growling vocals). We were digging the progressive intro of the opening track. Then came...the vocals. Didn't listen to them much after that.

With that said, I agree with much of what's been said already. A solid riff, a catchy keyboard part, an intriguing drum fill, or cool bass line will also get my attention. Although the lyrics may be the last thing I notice, I have to say that without good lyrics, a band will not be one of my favorites. The lyrics are what draw me deeper into Theocracy's excellent music. In fact, another solid release or so, and they may overthrow Dream Theater as my favorite band. And Dream Theater has been my favorite band for almost 20 years!
 
The first things I notice are the overall sound and feel (melodies, rythm, tone, mixing), and the catchy/original/interesting parts.

After some listens I start to look for details : it may be keys that are put at a very low volume (there are a lot of these in Theocracy), funny things on drums (like playing with the ride or the double bass :D), or on guitars (for exemple at the end of I AM "I am the joy of angels dancing in the streets of heaven" there is a pretty cool riff), or even on bass sometimes...

I must admit that the production is very important at the first listen (When I discovered Theocracy I didn't like the first album because of the production, but I gave it a chance later and I realized it was a bomb), and the vocals too : a song without vocals is better than a song with lame vocals IMO...
 
For secular music -- cool guitar stuff and lyrics that are not patently opposed to a Christian world view.

For Christian music - good melody, sound, and chord structure will catch my ear at first, but once the lyrics become apparent (sometimes it takes a few listens to tease them out of the mix), they become the mainstay...if a song doesn't say much, or, worse, does not line up with Biblical truths, I quickly set it aside.

---Prem
 
Gonna be blunt here. Technicality or catchiness. If I can tell it's technical right away, instant win for me. If not, it's gotta be catchy enough to grab my attention and go, "hey I like this."

Vocals never really bother me that much like a lot of people in this thread are saying. Unless they are poor vocals on power metal...which most are.
 
I'd have to say that the overall quality of the music's composition is what I find most appealing. When all of the various parts of a song can come together in a way that fits, I am happy.