I second Mullholland Dr. That will keep y’all talking for weeks.
How about Greenaway’s The Pillow Book?
Excellent choice - a cartoon fairy tale set against the Red Scare of the late '50’s but so much more. You can see why Pixar snapped up Brad Bird (the director of it, The Incredibles and Ratatouille) from Warner’s as soon as he left when he was disappointed with how The Iron Giant was (not) marketed. Brilliant movie.
I’ll second Gattaca, The Truman Show, Color Purple and The Manchurain Candidate (old version).
Unforgiven Clint Eastwood’s Oscar winning western. (Just what does the title mean)
A Perfect World Clint Eastwood again but a little seen but really great movie.
No Way Out (Ask, just how pissed off were you at the ending?)
My Life as a Dog from Sweeden, directed by Lasse Hallstrom (who also directed The Cider House Rules, and What’s Eating Gilbert Grape)
oh and What’s Eating Gilbert Grape would be good too.
Cinema Paridiso from Italy, should be easily rentable.
One of my favorite movies Empire of the Sun It is really a very visual film. Telling the story though mostly images and not relying on dialogue.
OH and Citizen Kane
I’ll second (or is it third?) Memento, Casablanca and The Iron Giant, all of which are simply excellent.
I’ll also nominate The Incredibles (exciting and fun, wonderful eye-candy, with a sly take on midlife crises, modern marriage and child-rearing), Dark City (a creepy, noir-ish, well-crafted and ultimately exhilarating sci-fi tale), Trading Places, The Big Lebowski and Animal House (just about the most perfect comedies ever, IMHO), as well as In the Line of Fire (an exciting political thriller, with Clint Eastwood at his best as an aging Secret Service agent still guilt-wracked over his failure to save JFK), and last but far from least Breaker Morant (a powerful, well-acted, stunning courtroom drama/miltary adventure).
This is an excellent list - at least of movies that I want to see. 
There’s a lot of good films mentioned - thanks so far, everybody! Keep 'em coming - we’ll be able to mine this thread for a couple of years. 
I’m dead serious. The theme of both is “Family is where you find it.” Neither protagonist can function in the family he was born into, or without the family he came to find.
I’m going to give you some…different choices.
Oldboy (Korea, 2003) - a film about revenge, taken to extremes.
Audition (Japan, 1999) - Again, revenge of a sort, but this one’s trickier, and could make for an interesting analysis.
Kwaidan (Japan, 1964) - four ghost stories, one incomplete. Beautiful film, good for discussion, but long and very slowly paced.
Man Bites Dog (France, 1992) - a fake documentary wherein a film crew follows a serial killer. May make you very uncomfortable.
And if you really want to make people uncomfortable: Cannibal Holocaust (Italy, 1980) - a film crew disappears in the Amazon. When their lost film is discovered, it shows savage and horrifying events that unfolded when the documentary makers found a lost tribe. Warning: the recent restored version of the film includes scenes of actual animal killings that are quite graphic. There is an option in the DVD menu to watch the movie without those scenes if you wish.
The Beast (originally The Beast of War). A thinking person’s movie about a tank. Sort of.
Sailboat
The Wild Bunch
Unforgiven
Tombstone
Once Upon a Time in the West
Twelve Monkeys - Was Cole really a time traveler or just a crazy person who imagined he was a time traveler? Or a crazy person who was a time traveler? Was he affecting the course of events or just being affected by them? Do the apparent contradictions in the movie come together to reveal one truth? What does the ending really mean?
I’ll second Brick, Casablanca, and The Incredibles.
I’d like to add Sleeping Beauty (Disney version 1959). It’s not perfect, but it has the best visual style of any Disney picture and is simply beautiful to watch.
If you go with Twelve Monkeys, you’ve got to add La Jetée.
If you go with Forbidden Planet, you should also watch This Island Earth. (Well, maybe.)
American Beauty comes to mind…
Why, in Metaluna’s name? (asked CalMeacham)
Read the book. Ignore the movie. Unless it’s the MST3K version.
I second Dark City and Twelve Monkeys, and add Artificial Intelligence and 2001. (All three will certainly bring debate / discussion. Sci-Fi worth watching.)
Some “Thinking-Man’s Action Flicks”
Ronin, Heat, and i-Robot.
Okay, I can kind of see your point, but I think it’s a bit of a stretch with “Boogie Nights”. Yes, the “family” aspect is there, but I see much more “drugs lead to self-destruction”. Plus there were so many other side stories in “Boogie Nights” that were more off the “family” theme.
“My Life as a Dog” was clearly more about “finding family”, but also about surviving/adapting when life throws you shit.
Back to the OP:
I 2nd (3rd, 4th ?) “Unforgiven” as well. Don’t know how I overlooked that one.
As well as 2nd “Donnie Darko”
One of my favorite movies for movie buffs that always gets overlooked is
Two Lane Blacktop
It’s one of the very few movies that does existentialism without pointless and stupid Surreality
As a contrast, perhaps. I love TIE a lot, but there’s no denying that FP is by far the better movie.
However, TIE does a wonderful job of gradual escalation: Things start out mundane, get a little wilder, then wilder still, and by the end you’re watching aliens bombard a planet with asteroids–and it’s still believable, because they cranked up the heat so slowly.
And it’s got Faith Domergue, who is one of the strangest-looking beautiful women I’ve ever seen. And square-jawed studly Rex Reason. And Jeff Morrow, who appeared in just about every '50s sci-fi movie ever made, it seems like.
OK, OK.
This Island Earth was shown to me in a sci-fi cinema class in college. The film was rushed out by Universal to beat MGM’s Forbidden Planet, but was, and still is, really bad, especially compared to FP. TIE was the original source of the Bug-Eyed Monster and the inspiration (I believe) for the Coneheads, but will always be remembered as the other big-studio sci-fi flick of the 50’s (ya’know… the bad one). I’ll always have a fondness for it.
My storyboarding instructor always said, from a filmmaker’s perspective at least, you should often watch terrible movies to see what not to do. (He then proceeded to narrate the errors throughout the entirety of Plan 9.)
In a slightly unrelated question, the SDMB should have a movie watch-and-discuss series like the wine club. Anyone interested?
bolding mine:
I think that would be a lot of fun.
My Life as a Dog and Boogie Nights? Well, I walked out of Boogie Nights and My Life as a Dog is one of my favorite films.
However a great discussion point for Dog would be its treatment of sex. Specifically kids and sex. The kids in US movies are basically sexless, and not interested in sex or discuss sex ever. In MLaaD, the kids do. It is done in a very realistic and tasteful way in the film.