Grave of the Fireflies is a good one and has been mentioned a few times already.
Fox and the Hound has several tear jerker moments, though most memorable for me is when Todd is returned to the forest.
Land Before Time was my first parent death in a film that I can remember. And unlike Bambi, this movie shows more of the consequences of losing a mother at that young of an age.
It is a cinematic triumph and a film that everyone should see. With that said, I am unable to ever view it again. The scene where Schindler is breaking down realizing that if he didn't have expensive items he could have saved more people just absolutely killed me.
Just saw Elephant Man at the Music Box Theatre. Incredibly moving film if you haven't seen it. They're currently running a film series on David Lynch and that was the first of his films I had ever seen. Planning to see more this week if I can!
"Paths of Glory" is one of Kubrick's most underated films (or at least lesser known) and the ending is pure emotional power.
The fact that you just spent almost 90 minutes hating humanity and the shitty situations in which we put ourselves as a species, only to be able to come away from the film thinking "we're not all bad" because of what happens in the final 10 minutes blows my mind.
If you haven't seen it, I can't recommend it enough.
An Affair to Remember (1957)
Yes I did watch it because it is referenced in Sleepless in Seattle and yes I did blubber cry just like the SiS characters suggested.