The first movie I remember uncontrollably sobbing during was Steel Magnolias. My great grandmother had passed away a few years prior to watching it. Death was something my family always tried to shield me from, so they were never honest with me about things that happened when people passed away. The day my great grandmother passed away, I remember hearing someone say "They unplugged the machines". I was around 9 at the time and didn't understand what that meant, I thought someone had accidentally unplugged the machines and killed her. When they turned off the machines in Steel Magnolias, I finally understood what happened to my great grandmother and it was deeply upsetting to me, but it also helped me heal.
City of Angels. I didn't know that it was sad, the commercials made it seem so happy and lovey-dovey. And this was way before I would research movies at all, so when Meg Ryan's character died, I was gutted!
Titanic, of course. I sobbed so much at the end of that movie that I couldn't leave the theater for about 10 minutes after the movie was over. The same thing happened when I saw Armageddon in the theater. But that was because my boyfriend at the time had just left for the Marines.
The Notebook kills me emotionally, too. The first time I watched it I was realizing my first marriage was over, and it hurt bad. But I watched it many many years later with my DD and it still kills me, even when I know what is going to happen.