When I was a kid it was Old Yeller (if your eyes don't at least get misty when he has to shoot Yeller, you don't have a heart)
Later it was Ordinary People. Mary Tyler Moore's character was dead on to my mother; no tears but it helped me see my family for what they were
Then it was a little film called "Lady Jane" with Helena Bonham Carter and Cary Elwes about Lady Jane Gray; I saw this a week after I was released from a psychiatric hospital. I didn't cry but the film, particularly the ending moved me deeply.
But the one that hit the most, and stayed with me the longest was a made-for-TV movie called "I Know My First Name is Stephen" about the real life abduction of Stephen Staynor. No tears I identified with much of the story and I was truly saddened when I heard he died in a motorcyle accident.
However the one scene in a movie that made me cry more than any other was from, of all things, Titanic. When the ship was sinking and they came upon an Eastern European boy standing in front of a door barely holding the sea back. I damn near lost it in the theater.