I am an undergraduate mathematics major and my cultural background is mostly American. That's all I have to say about myself, really.
More importantly, a bit about my tastes. I'm not really an anime fan in the sense that I like anime any more than conventional television or film. Like any other storytelling medium, there is some good anime but a lot not worth watching at all. I don't view a series relative to only anime series, but in the context of all television (animated or not).
So, why do I watch anime? Well, one thing I do like about anime is its format. I like the idea that pretty much all shows end after one season. You get a greater variety of shows, they're much more manageable in terms of time, and you don't have as many shows that have a great first season and then milk the franchise for another ten. That having been said, there are a lot of common traits of anime that sort of annoy me. I won't go into all of them, but the biggest one is what Alfonso Cuaron described as "expository film," of which he said, "There's a kind of cinema I detest, which is a cinema that is about exposition and explanations.... It's become now what I call a medium for lazy readers." What does this mean exactly? A lot of anime are have plots that are solely focused on explanations; mysteries are presented and they're explained. A lot of anime also lack even barely acceptable dialogue, one of the most basic yet greatest narrative tools in cinema. Another example is when the characters' internal dialogue is voiced in order to portray their emotions or illustrate their thought processes. This is lazy, both for the writer and the reader; the writer doesn't have to think of a creative way to portray their character, and the reader is saved from having to expend the effort of actually reading between the lines. While exposition is sometimes necessary, it is a weaker narration; relying too much on exposition, no matter how "intriguing" the plot, will leave people bored.
Another aspect of anime that I'm not a huge fan of is the exaggerated importance of plot and philosophical/moral meaning. First, after you've read enough and watched enough material, you realize that most plots are really just variants of each other; to say a plot is "original" usually means you just haven't seen that other series with a similar plot (it was said that Shakespeare was the greatest plagiarizer). There are numerous examples of mundane plots turned into great work and examples of very "creative" plots that fade into obscurity. Second, most anime are not philosophically groundbreaking; they take that material from various sources (most commonly, existentialist works). While I don't necessarily oppose the presence of philosophical themes in anime, they alone do not make a work great. If I've read Being and Nothingness, what can I possibly gain from a series that only rehashes the same material? A series must offer something in addition. However, a philosophical or moral message that I find offensive or absurd can easily cause me to dislike a series.
I'm quick to drop series. I don't like wasting my time, and you can usually roughly tell how you're going to like a series within the first few episodes anyway. If I absolutely hate a series after the first four episodes, why should I subject myself to watching any more? Unfortunately, to do a proper review, one should watch the entirety of a work. Thus, I probably won't be writing too many negative reviews, even though I'm so full of negative thoughts.
I'm also not too big on shounen anime in general. I'm also not really into the weirder genres like yaoi or whatever.
It seems that the most popular way to do an anime review is to rate a show based on plot, characters, animation, and sound. I don't like this because it's saying that somehow these are four fundamental aspects of television. It also loosely implies that all four are necessary for a good show. I think this is too rigid a characterization. First, I don't think all are necessary (in the spirit of modernism), nor do they all have equal weight, and deficiencies in one area can be made up for excellence in another. Furthermore, it implies that excellence in these four categories is sufficient for a good show, which is equally preposterous. Reviewing is a subjective endeavor, and I treat it as such. Thus, my reviews (if I ever write one) will try to determine the essence of a series, and how it was executed, rather than rely on some fixed formula for writing reviews.
There are some phrases that reviewers overuse when they don't really know what they're trying to say.
1) "thought provoking", "deep"2) "well-developed", "three-dimensional" [characters] - Having "three-dimensional" characters is not a characteristic to boast of, because it's almost considered necessary. On the other hand, having two-dimensional characters is something to mention.
I really hate Death Note. I think it's one of the most intellectually insulting shows I've ever seen. Just wanted to put that out there.