The first category I use (and probably the most important of all) is the story concept and how the plot progresses. A great story begins with an interesting premise. In addition to having an imaginative story, a good story teller needs to include elements like secrets, big revelations and sudden changes to keep the audience interested.
The second crucial element in a great anime is character development. For me great characters are those with complex issues and/or unique personalities. I greatly admire anime creators who are capable of making us fall in love with the characters. This can also be done by showing us how the characters grow/mature overtime. In addition, it helps when we can relate to the fictional characters.
The third most important aspect to determinate the quality or greatness of an anime is how the story concludes and whether or not it makes sense. A great conclusion is one that tries to tie all the loose ends and explains what is going to happen to the main characters. Logic for me is when the different elements introduced in a series lead somewhere. A bad conclusion can downgrade an anime.
The originality and uniqueness in a series is not a make or break element in an anime, but it can enhance the experience. Anime, like any other form of media, often borrows elements from past works. As more anime is created, it is hard to come up with something 100% original. If an entirely new concept cannot be developed, great anime makers can use their ingenuity to create unique experiences based on existent concepts.
The last, but not least, element is production values. This mainly refers to the quality of the animation in a particular series. But it also deals with things like what kind of "camera" angles are used, the sequence in which events and episodes are presented and how effective time is used when the series are short.
This week I want to used high profile anime series to better illustrate my points. Be warned, it will contain some spoilers.