Characters - For me, characters are the most important part of any anime. I need to be able to enjoy the characters for me to really like the anime. An anime could have great animation, sound, and story, but if I do not like, or even hate, the characters, I probably won't give it anything higher than a 7. Characters are what breath life into an anime, and they are usually what keep me most engaged in an anime.
Story - Story is also really important for me. Obviously I want to watch an anime that has me on the edge of my seat with a unique or well written narrative. However, I am also willing to except generic/plain stories so long as the characters are well-written or engaging enough to keep me interested. In some cases I would say that the story is just as important as the characters, if not more so, but normally characters ranks higher than story for me. I also want to point out that it is hard for me to watch shows that are for the most part plot-less, such as SKET Dance and Gintama, so I do look for some sort of story to follow.
Animation - Animation is important, because anime is a visual medium for the most part, but it is also telling a story about the characters. I do appreciate good animation, and it is hard for me to watch some anime with either bad animation and/or extremely unique animation, but for the most part I'm not too picky, and there are very few anime I didn't watch because of animation. I'm able to handle anything from the really crisp new animation to the old school animation. I do consider it important, but not as important as characters and story.
Sound - I more or less consider sound a bonus. If it has a soundtrack I really like then that's great, but it isn't going to get extra points for that. The same goes for a bad soundtrack, if it has a soundtrack I don't like, I'm not going to knock off points for that. Then there are things like background noises and character voices, I also consider these to be bonuses. If they sound good that's awesome, if they don't then oh well. For me to add a point for sound then the voice acting and soundtrack would basically have to be out of this world, and for me to take off points for sound it would have to be pretty terrible.
ENTERTAINMENT - Some people rate anime based off of entertainment and some don't, I am the former. While I can respect people who do not rate based off of anime, and I can see where they are coming from in their reasons not to do so, I just wanted to let you all know I do rate factoring in entertainment. The way I see it, at the end of the day an anime's purpose is to entertain the viewer with an enjoyable watching experience, and therefore it should be factored in.
A simple summary of what my scores mean:
9-10 = A
7-8 = B
5-6 = C
3-4 = D
1-2 = F
My current mean is basically a 6.6, so out of the anime I have seen, I basically think the average anime is between a C+ to B-
A slightly more in depth look at the scores I give.
10: Masterpiece - Basically something that I enjoyed a lot and had very little flaws that were overshadowed by all the things I liked about the anime.
9: Great - Similar to a 10, but with either just a little bit more flaws, or the anime had less impact on me in comparison. Still an amazing anime that I consider a favorite.
8: Very Good - Something I really enjoyed, but nothing that blew me away.
7: Good - Simply put, something I thought was good. Had some flaws and/or qualities that held it back in some ways, but overall still a good experience.
6: Above Average - An anime where there is slightly more pros than cons, but the cons are abundant or relevant enough to hold it back from being a more enjoyable experience.
(Hypothetical 5.5: Average - On a 1-10 scale the average score is actually a 5.5, but MAL does not allow us to give that score. To me this would be a show that is 50/50 in terms of pros and cons, but for me usually the pros outweigh the cons or vice versa. I have yet to watch an anime I have felt 100% 50/50 on).
5: Below Average - Sort of like 6, but where there are slightly more cons than pros, which still make it somewhat watchable, but hold it back from truly being enjoyable.
4: Bad - The cons outweigh the pros enough for this anime to be consider a bad experience. This is the lowest level of what I consider tolerable to watch.
3: Very Bad - This anime has a lot of things going against it. Has very little redeeming qualities.
2: Horrible - This anime was just not enjoyable in pretty much every way. Way too many flaws, and pretty much no pros.
1: Appalling - This anime would basically have to be 100% trash for me to even consider giving it a 1, with absolutely no redeeming qualities whatsoever.