The entire movie Grave Of The Fireflies is the saddest and most depressing anime film I have ever seen, but it at the same time beautiful. I recommend it to any fan of anime that has not yet seen it.

In Grave of the Fireflies when Setsuko dies. That was sad, I mean really, alittle girl died of malnourishment in the hands of her brother who was trying his best to do something during World War II in Kobe.

The only time I actually cried while watching an anime show was when Frieza killed Vegeta in Dragon Ball Z.  Then Vegeta spilled his guts out to Goku about the saiyan race and how it could be destroyed if he couldn't defeat Frieza.  Then Vegeta actually started crying himself.

End of death note.
 Or when Vegeta dies. I put it as second because he can come back to life.

I have 2.
Death note ending or when Ash turns to stone.

I think, the biggest time I really broke down in tears was during the anime "Fullmetal Alchemist" (SPOILER ALERT!)

It was during the time that a father (who is also a alchemist scientist) was doing experiments on people, and ended up experimenting on his own daughter... turning her into a chimera and then she finally got killed by another Alchemist who is like... extremely messed up in the head.

Just seeing that man turn his own daughter into a monster suddenly brought me into a state of rage, then seeing Edward Elric cry... caused me to cry, and I was like "WHY WOULD YOU DO THAT TO YOUR OWN DAUGHTER!?!? WHAT IS WRONG WITH YOU!?!?" and well... let's just say I broke down in a sobbing rage and ended up punching my floor until my knuckles were hurting severely.

==================================

But the very first time I ever cried was during the movie "Pokemon The First Movie: Mewtwo Strikes Back" and it was when Ash Ketchum got turned to stone and Pikachu was crying... and I was like "Not going to cry... not going to cry... not going to... PIKACHU, WHY!?!?" and well I cried my eyes out all over the place and that lasted a good 30 minutes.

Every time I see that freaking scene, I cannot stop myself from crying... it's just initial reaction, simply because I LOVE Pokemon and I can see how much Pikachu truly cares for his trainer.

Awwww, crap... just thinking about it right now is causing me to cry.

I haven't seen Clannad yet, so I won't be listing that. Grave of the Fire Fly's for me though. The entire movie is depressing.

SPOILER ALERT! DON'T READ UNLESS YOU WANT CLANNAD TO BE SPOILED FOR YOU!

Mine was when Nagisa Furukawa in Clannad dies, and then later on in the After Story, Tomoya and Nagisa's daughter, Ushio, dies. (I had it spoiled for me so watching it through right now as we speak. On the fourth episode of the After Story). 
 

Lelouch VI Britania's death in code geass

I would have to say that the saddest anime moment for me was in Death Note. The only time I can remember a tear coming to my eye was when Rem killed L Lawliet. ;-;

Oh my God, Clannad for sure.
When Nagisa dies after giving birth and the spiral Tomoya goes through and finally when he reunites with Ushio. Just heart wrenching.