Nice idea for a thread, reed.
I'm sure there are quite a few films I've seen that have got me thinking to various degrees and/or have sparked discussion with people afterwards. I'll name a few that I can think of off the top of my head:
American History X - I felt this was an interesting look at how people's attitudes towards others can be shaped, whether for better or for worse.
Bowling for Columbine - I know Michael Moore divides people, but I felt this started to address questions about fear, gun usage, and how the behavior of people in different nations can vary so much. I'd always felt the witchhunts against video games or particular films were simplistic.
A Clockwork Orange - A great film with a quirky environment (language, backdrop, music, etc.) and which addresses concepts of morality and "correction."
Come and See - This Belarussian film has its "arty" moments, but it provides a very different take on World War II. Instead of the usual, Hollywood-based depictions about the war in the west, we see what it was like in the Soviet Union and where the civilian populations were extremely fearful, being treated as sub-human by the advancing invaders.
No Man's Land - A Bosnian comedy(-ish!) that has the audience despairing at points, it paints a picture of where two soldiers from opposite sides end up being forced to spend time in one another's company, while the United Nations appear ineffectual and have to play to the world's press, who in turn enjoy the theater of it all.
Waltz with Bashir - An Israeli animation about an Israeli in modern times who confronts the memories of his involvement in the Lebanon War and Israeli complicity in the slaughter of Palestinian refugees at Sabra and Shatila. I actually found the animated, majority of the film more powerful than the ending, which showed real footage of the aftermath. This is perhaps because I am, unfortunately, accustomed to seeing footage of tragedies on the news, and so seeing an unusual format had a greater impact.
Well, that's a few to start with!