I'm the sort of person who is very easily moved to tears and someone who has watched quite a bit of anime in her time, so to list all of them that I've cried during would be quite a hassle. One particularly strange time an anime that made me cry, though, was Minami-ke. If you've never seen it Minami-ke is a humor/slice-of-life anime, by definition a generally very laid-back genre. But, there was a character named Fuyuki who I really felt I could understand, so every time things didn't work out for him I got very teary-eyed.
He was just this kid who got moved around by a single-parent so often that he never got to become part of a group or ever even learned how to be and so he just ended up sitting there quietly and doing whatever people asked him to because he didn't know how to say "no". So, basically me when I was a kid. Or me now. And probably me in the future. I'm pretty certain I'll be a socially awkward door-mat for the rest of my life. [/wallowing in self-pity and pity for fictional character]