It's really hard to say, especially as it can be dependent on the type of music. Ultimately, I like my music to have a meaning or purpose, rather than the mindless rubbish popular on radio, and it's no secret that I am especially keen on atmospheric, ethereal music that creates sweeping soundscapes. I very much also look for talented musicians, people who can actually play their instruments with both supreme skill and a great sense of tastefulness. That's why I really enjoy Steven Wilson and John Petrucci but find Yngwie Malmsteen really boring.
One thing that's very important to me is that if a band has a vocalist, the vocalist has to actually appeal to me, either as a great vocalist (Steven Wilson, Neil Finn), or as someone whose vocals fit the style of the music (e.g. most death growlers). One reason why I really strongly dislike Radiohead is because Thom Yorke is one of the worst vocalists I've ever heard - every slower song he sings resembles three cats being strangled to death.
I could go on and on, but I won't. It just baffles me why a mediocre band with an unimpressive debut like The Arcade Fire are the darlings of this forum while a band like Pure Reason Revolution releases a very accomplished, mature debut full of rich atmospheric soundscapes and I'm the only bloody person on this forum even talking about them!