Well for me, it has to be the atmosphere and emotions.
The characters, story and music are also really important, and if any of them are bad, than it could be really annoying (Although it's hard to mess up on the music department, unless they put music that doesn't fit in the least.)
Though I do have to say that the animation is pretty much secondary, although if the animation is inconsistent and doesn't look good, than it could still get in your way quite heavily.

Characters I'll have to honestly say are one of the main points of the anime. For example I find the tsundere's, who hit the main character into the sky for ''comic relief'' (*cough*lovehina*cough*) to be completely EXPERIENCE BREAKING. If there's a main heroine who is an aggressive tsundere, than I'll stop watching the anime in a heartbeat.

The romance animes can be pretty good, if they don't overdo it with fanservice, over dramatication (Where the girl for some reason runs out after doing an idiotic ''misunderstanding''.) or the or scene's which shout ''she's soo kawaaaiiii'' (Which is sadly quite a lot of animes.)

Basically I watch animes cause of the atmosphere and emotions, and I hate animes that will take the easy way out instead of making an anime that gives a great atmosphere without f-ing it up (Love hina, I'm still talking about you.. Though there's so many more of that kinda animes.)

A Harem where the main character(Boy) isnt a spinless whimp.

Generally it would be likable characters, uniqueness, and a story that actually makes me think about stuff. Music is somewhat important because I wouldn't call a show great if it's music was bad. Animation is the least important to me (this sort of depends on the show though). Death Note and GTO both are shows that fit this criteria (they were the last shows I watched). Pretty much a show that almost everyone could watch and like.

The plot and Characters. If i don't care about what the main idea is then i probable don't care about the characters which means that every Suspenseful scene will just have me not caring if the characters live or die. Once i like the characters enough to care about the safety then i can watch it regardless of the art quality/style "Video girl A.I" is one of the ugliest things i've ever seen, but the characters interested me enough to keep watching.

EDIT: I think i just talked in circles, but Characters are the most important part

In order:

Character Depth
Dialogue/Character Interaction
Music/Atmosphere
Fascinating Story, World, Characters, etc.
Art Style

I don't really care much about having really good or likeable characters, but it is important to me that what characters there are have some depth to them. For example, Menma from Anohana was annoying, boring, and generally uninteresting. However, her character was explored very thoroughly, and as a result, you understood her and the connections that everyone had with her. To compare a similar show, Waiting in the Summer had characters that were far more interesting and, overall, more enjoyable as characters. But by the end of the show, you don't feel nearly as close to them. So you can have great characters, or boring characters, but the exploration of those characters is far more important to me.

Dialogue is pretty self-explanatory -- I like hearing people talking about cool things, and if possible, doing so in a cool way. How the characters communicate outside of speech (making a face, body language, actions, etc.) is equal to dialogue as far as importance. Bakemonogatari is probably the standard here, although I've seen a lot of others that may be better on close comparison. V for Vendetta or Thank you for Smoking are both great examples outside of anime. God, I love the talking in those movies.

Music goes a long way to helping me like a show, and is probably the biggest contributor to the atmosphere, but really what I care about is the feeling/ atmosphere.

Being fascinated or interested in something is also something to hook me if all else fails. Eva is a perfect example of this, as I was not impressed in any respect as far as the previous 3 elements were concerned, but put simply -- the show is fucking interesting. Shin Sekai Yori is a recent show that I'm watching almost exclusively because I want to learn more about it. It's purely fascinating, but not really enjoyable otherwise.

Art Style is the only other thing that contributes heavily to my enjoyment of a show, and it's pretty self-explanatory. Although animation quality is important, the actual artwork is much more enjoyable to behold when done right. Anything by SHAFT, Black Rock Shooter, Gankutsuou, Mushi-shi, and more recently, Shin Sekai Yori are all shows that I just love to look at. Whereas something like Haruhi, or more recently, Vividred Operation, which have really fluid motions and just a high-quality finish, aren't really as interesting aesthetically to me.

That's it. Humor, well-made characters, specific story elements, uniqueness, and almost anything else -- while nice to have -- are really sort of secondary or derivative to the above elements. Cool topic, even though I'm almost sure I've seen it before.

There really isn't just one thing for me but I can say this; when I'm watching it and all of a sudden it's over and I'm like no that couldn't have been a whole episode. So I guess it's when the story really pulls me in and I become emotionally invested in the characters and whatever they are trying to accomplish.

What does OP think about this topic?

To me, a great anime is one that has you thinking about it even after it's over.

Story, sound, and art... interesting must be interesting....

Forgot to answer my own question lol

IMO to have an anime be considered great by me, it would have to be the characters that really pull me in
Cookie cutter characters do nothing to wow me unless the approach/formula to that character (background, history, actions, dialogue, etc.) is different
It's a plus if the characters are somewhat relatable, that you can associate with on some level that you can understand their motivation/feelings to do something or another (this isn't the case in some anime I think are great that are fantasy based which is why this is just a nice bonus if the characters are like this)
Characters really are the things that move the plot for me, the plot can be something as generic as saving the world but if the characters are great, then the plot can be whatever it wants

My example: Arakawa Under the Bridge
I personally think its a great anime, but is there really a plot? IMO not really in a sense (the whole Nino Venus thing is kind of meh and more of a background overall plot but you rarely notice it, so you can say the story plot isn't strong) but the characters are enough to "wow" me anyway

Great characters or a great plot/story, of course I'll prefer both if possible.

I don't honestly have a definite line between "good" and "bad", let alone "good" and "great". There's just way too much to factor in, especially when every show is in some ways different with different genres and standards to judge.
I suppose the very basics of what I usually desire in animation is striking visuals, entertainment value, and deep stories with well thought out characters. However, it's safe to assume that's what a lot of people look for in any visual medium which tells a story, so I guess my input is nothing special.

Really depends how bored someone is, I'd have to say it depends on how good series are beginning to end. . everyone eventually turns to naruto / bleach they're OKAY but there's better XD

most of the good animes aren't really hyped up anymore but if you hunt around you'll find really neat ones X)

characters o_o

-funny/silly
-clumsy/carefree

dunno what's 'great' there's sooo many anime I love!!

What makes a great anime is variable... but most commonly it comes down to engagement. How much do I feel attached and involved in what is happening. The questions I ask myself are typically:
Do I really care what happens?
Do I have strong feelings, one way or another, for any of the characters?
Do I spend my time when not watching it thinking about it?
Do I ever find myself going 'holy crap I can't believe that happened'?

If the answers to these questions are more often than not yes... then I consider it at least a good anime.

I also use the same basic rubrik I use for books... a good anime I can enjoy watching, a great anime I want to watch again.