Here are some films I'd classify as first-tier films (Tarkovsky is one level above "first tier", though some -- perhaps just a few -- of the ones below might also rank "above first tier"). If one or two films snuck into this category which shouldn't have, because it's been too long since I've seen them, I apologize (to myself of course, since this is just the list of my personal preferences, not someone elses).
Bergman's The Magician (also, I seem to recall, Persona); My Left Foot (film based on the true story of a quadriplegic); Black to the Promised Land (Directed by Madeleine Ali, 1992, Israel. A documentary about a group of black inner city kids are taken to an Israeli Kibbutz by their Jewish teacher); Zero For conduct (France, 1933), Europe Europa (Based on the true story of a Jewish teenager in Nazi Germany who survives by posting as a Hitler Youth in an elite Nazi Academy. Directed by Agnieszka Holland), Orson Wells' The Trial (based on Kafka); The Brother from Another Planet (One of many low-budget, independent, fresh, and thoughtful films I saw at Cornell. This one on a black "alien" visitor to Earth); A night On Earth four (five?) scenes of and about human life at four places, all occurring the same night, around the world;
The Nasty Girl (Germany, 1990. Title sounds erotic, but in fact, is includes romance and daily trials and tribulations from the young woman's point of view, with a more major plot developing into her discovering her small quiet little town in fact harbors some "secrets" from the Nazi past); Pink Floyd, The Wall and the quite unrelated but riveting The Wall (Turkey, 1982, on brutality in prison and society; Frida (Mexico, 1987 on the artist); Man of Flowers; and on sexual themes, the freshly sex positive Annie Sprinkle's Sluts and goddesses and the surprisingly sensitive and poignant The Masseuse, which belongs in that 1% the "Adult Film Industry" can be proud of. Wings of Desire (thought the sequel was somewhat disappointing); Leolo
These are either "second tier" (but still good) or else films I've seen too long ago to remember well how strong an effect they had on me. Brazil, Sammy and Rosie Get Laid; Peggy Sue God Married and another film playing with time, Groundhog Day (but not "back to the future", which was entertaining but not on my list of Good films); My Dinner with Andre; Swamp Thing; Volere Volare; Sugarbaby (a German film in which a middle aged, overweight woman gets the man she lusts after, via her extended complex efforts -- though I am unattracted to obese women (she was not just overweight, but obese) the film was done with such sensitivity and humor that you root for her :-); the only truly scary film I liked though wouldn't want to see it again, The Vanishing (the foreign version), ; Metropolis.
Also The Return of Martin Guerre (based on a true story; there was another film, also based on a true story, also from Europe, and which I saw at the same time whose name I can't remember, where a man is found who is an adult who grew up outside of society, who tries to learn their customs, etc. At one point the two-robots-dilemma is presented to him, one robot always lies, one always tells the truth, to which he replies, "I would ask one of the robots if he were a frog" :-); Tampopo, Diva; The Last Emperor; 2010 (which unlike 2001 made a reasonable amount of sense); 1984; and Harold and Maude. Also Dead Poets Society and Nosferatu (the original vampire film); Blue Velvet though I'm iffy, based on what I recall if it's deserving of second tier...; the original Wizard of Oz; and some of Woody Allen's "serious" films. Oh yes, and Sigorney Weaver in Alien 3 for some humanity, mostly UNglorified violence, psychological drama, feminism in a positive (see "UNglorified violence", above) sense, not to mention I found her sexy ;-)
Some semi-political (Plastic Jesus from Yugoslavia) and political films I liked include Warrior: The Life of Leonard Peltier and of course, Manufacturing Consent: Noam Chomsky and the Media).
Movies I've heard good things about or am not sure if I saw them or not(!), and two (Clockwork Orange and One Flew over the Cookoo's Nest that for various reasons I only saw parts of but not the whole... Death by Hanging (Japan, 1968, against the death penalty, an "expressionistic" and sometimes darkly humorous film); My Beautiful Laundrette; Grapes of Wrath; The Gods Must be Crazy; This is Spinal Tap and I've heard good things also about the independent low-budget, El Mariachi.
And on the lighter side, Time Bandits and some of Chaplin's and Marx Brother's films; a few rare good TV series from years past (Mary Tyler Moore Show and Get Smart) who had intelligence and heart (or at least humanity) in addition to humor. Some of Woody Allen's, (e.g. Sleeper, Zellig, etc, thought it's hard to remember "how good" they were).
The Airplane and Naked Gun movies; the World According to Garp (does that count as comedy?); Mel Brook's History of the World Part One (also Young Frankenstein), and Monty Python's Life of Brian and Holy Grail, and the original Pink Panther.