Eraserhead - One of David Lynch's best films. A surrealist nightmare about the fears of an unplanned pregnancy.
Gojira/Godzilla - The original classic using a monster as a metaphor for Hiroshima, and the dangers of nuclear weapons. Tense, dramatic, and with great use of special effects, I think the original is the best in the series.
M - Fritz Lang's masterpiece about a child murderer in Berlin and the effect of his crimes on the city. How different levels of people in a society react to a crisis, and also an analysis of the killer himself.
The Manchurian Candidate - A political thriller with a unique plot of brainwashed P.O.W.'s sent back to America, with one being a sleeper agent to perform an assassination.
Persona - Ingmar Bergman's mind fuck of a drama about a nurse caring for an actress who has gone mute, trying to figure out what has caused this. With both getting to know each other on a remote house by the beach, past trauma and current hardships are revealed.
Police Story - Some of the best stunts, fights, and action sequences you'll see in a Jackie Chan movie.
The Texas Chainsaw Massacre - Horror classic that even today feels all too real. Preying on the viewer's fear of being alone, isolated, and a stranger in a rural environment.
Wild Strawberries - Another Bergman classic. This one about an elderly man nearing death, looking back and analyzing his life. Touching, heartfelt, emotional, and relatable, this is a must see.