Chobits - First and only anime that ever made me cry. And of course Chii is so kawaii!
Anime make me cry all the time but I am an emotional guy (I cried at the end of "Saving Private Ryan"). Maybe it comes with age, maybe it has to do with knowing so much loss myself, maybe it is an empathy thing but likely it is just me being me.
Great anime that I have watched and is emotionally moving (in my personal opinion of course):
Little Busters! (that includes the sequel
"Refrain")
RahXephon
Toradora!
Sword Art Online (really, how can that story not move someone to the core?)
Planates
We Without Wings
Another (I don't know if this is a truly sad anime but it is not a happy one)
Unbreakable Machine Doll (mostly a great action and comedy anime but includes some seriously deep humanity issues )
Kaze No Stigma (another good action anime)
Beck (music based fantastic story that should draw a tear from the corner of your eye)
Welcome to The NHK (one of my all-time favorites, recently rewatched it)
The Book of Bantorra
Haganai (and the sequel)
Red Data Girl
Darker Than Black
Gunslinger Girl (this is just a special anime but if you have a heart, it will tear it out)
Kimi ga Nozomu Eien (this anime was made to break your heart!)
Say I Love You (all romance anime is filled with heartbreaking moments, if it is any good and this one is)
Angel Beats
Kids On The Slope (great jazz too!)
Of course not all these anime will bring you to tears but almost all anime include some heart-rending drama. Full-Metal Alchemist is a great example, just the basic plot is almost enough to tear you up. The cartoons I grew up watching and the anime that comes out of Japan are so foreign to each other as to be unrecongizable as the close cousins that they actually are. Action and fanservice are great but if you watch anime...be willing to be moved by the depth of the story. When you watch a great story you become emotionally invested and that is what enjoying great anime is all about.