"12 Angry Men" (1957) is a personal favorite that I recommend to pretty much everyone. Great messages about questioning assumptions, challenging biases, understanding the limitations of evidence, acknowledging imperfections in the justice system, and the consequences thereof.
The movie is also cinematically interesting to me because it feels "small". The entire movie just about takes place in one room, and the events of the film transpire over the course of one afternoon.
In the beginning of 12 Angry Men everything is shot from above eye-level with wide-angle lenses, giving everything the feel of more space, but as the film progresses it transitions to tighter shots with telephoto lenses from lower angles. The film gives the viewer more and more of a subconscious sense of tension and claustrophobia as the story progresses.
At least one stage adaptation of the story gave a similar effect over the course of the show by slowly tightening the lighting and having the walls of the set physically move inward, too slow for the audience to take notice but enough to subtly affect the entire atmosphere and really drive that feeling home.
I've seen this movie 3 times, each viewing several years apart. Each viewing solidified this movie as my personal favorite, I would go in thinking I've already seen this, it can't have the same wow factor. Nope, I would end up loving it even more. Can't recommend this enough!