I'm worst when I see The Land Before Time and The Lion King - for Littlefoot's mother and Mufasa. I didn't cry when I was really young, though. Only after I turned 11 or 12, and I really understood what the death of an older family member meant.
Other that get me are Watership Down, Tarzan (when the baby gorilla gets eaten), which I saw in the cinema. Worst birthday present ever for a little girl who loved animals.
I couldn't watch it for years afterwards. Oh, and the Snowman!
Films that made me tear up (but not cry completely):
The Iron Giant