In order of importance, from most to least:

1.) Characters. I cannot stress this enough: if I do not like the characters, I will not like the Anime. The best animes will have characters that click with me on an emotional level. Of course, if I hate a character on an emotional level, that will also be considered a good character.

2.) Plot. There is a very close margin between plot and characters for me. However, because many anime end up being weaker in terms of plot than characaters (if only slightly,) Plot ends up at number 2.

3.) Music. Music can have a major impact on my enjoyment of an anime, but it's not necessary. Naruto has my all time favorite anime soundtrack, but it's no where near my favorite.

4.) Animation. Animation isn't that important to me. However, if anything, it will keep me from watching something if all I have to go by are screen shots. For example, it took me quite some time to actually want to watch One Piece because the art style looked so awful.

5.) Fighting. Even in action based anime, if fighting is all there is (such as DBZ or Flame of Recca) I will not be impressed. In fact, too much fighting actually detracts from my opinion of anime unless the fighting is well done (such as it is in LastEXILE, Naruto and One Piece.) However, it remains the least important thing to me.