Thinking about it, it's the crying/screaming scenes about unobtainable things that seem to set me off, especially if there's a really strong narrative soundtrack behind the dialogue/action. Let me elaborate briefly and forewarning here - spoilers if you've not seen Laputa: Castle in the Sky (in Japanese), nor The Boy in The Striped Pyjamas.
Anyway, in Laputa there's a scene when a salvaged guardian robot goes on the rampage and Sheeta's pleas for it to stop fall on deaf (nee metallic) ears as it goes about destroying the military; mind you, Sheeta as a whole in that filme gets me all fatherly...
Moving on, in The Boy In The Striped Pyjamas, the final scene of the film where the Nazi family realise that their son has disappeared and disguised himself as a Jew in a concentration camp, just as he is about to be mistakenly showered... and the music goes silent with the mother screams in anguish, finding Bruno's clothes by the hole under the fence... and the camera pans away from the now-silent chamber door... chilling.
So yeah, stuff like that in films get me tearful - though only if the music and the voice-acting/acting are convincing enough to pull me in.
Oh and moments depicting despair - I can't say it makes me feel happy, but ... I don't know, it makes me feel satisfied somehow and I just like watching it (hence why I'm a huge fan of anime films like The End of Evangelion and Metropolis and also live-action films like The Piano, The Road and Dark Water)