The numbers I used above reflect when I judge something rather than how important it is.
Answering your question, though, I'll write what I think are the most important things in order:
1. Originality
>Is this series original? Are the characters unique? How often has the story been done before? This is the most important.
2. Story
>Putting originality aside, is the story interesting nonetheless? Is it fun? Quirky?
3. Art Originality
4. Art Quality
>These are both very important, though one is more important than the other. For example, let's look at
Maria + Holic. That art! I could care less about some nose-bleeding lesbian falling for some blonde with a dick (<-that story). Anyway, the art was extremely original as well as very high quality. I have, however, seen animes that have HQ art... but it's just not interesting. It's not original. That's why having original art is a lot more important to me rather than having HQ art.
Let's look at some other animes.
Interesting and quality art:
Maria + Holic,
Bakemonogatari, Ergo Proxy, Redline
Interesting but lazy art:
Sayonara Zetsubou Sensei,
The Count of Monte CristoExpected but quality art:
Munto TV,
Nyan Koi,
The World God Only KnowsExpected and lazy art: Naruto, Bleach, The Melancholy of Haruhi Suzumiya
Middle of the road: One Piece, Naruto Shippuden, Kuroko no Basuke, and a lot of others that I'm too lazy to list.
Anyway, you get my point. Moving on.
5. Characters
>Are the characters realistic? Original? If not, I probably won't enjoy the story as much as I want to.
The rest of it all kind of melds together. As much as I appreciate good animation, I would never refuse to watch an anime for its animation (especially if it's an older one). The same goes for the rest of the points that I brought up earlier; if it has a bad sound track, it's not going to make me stop watching it, so it's not really important enough to bring up here.