In no particular order:

Almost anything by the Coen Brothers, particularly The Hudsucker Proxy, Fargo, The Big Lebowski, O Brother Where Art Thou, No Country for Old Men, and True Grit.
Almost anything by Martin Scorsese, particularly Taxi Driver, Raging Bull, The Color of Money (though watch The Hustler first), Goodfellas, Casino, The Aviator, and The Departed.
Moon
The Mission
12 Monkeys
A few of the David Fincher films, particularly The Game, Fight Club, and The Social Network.
The Lives of Others
Little Miss Sunshine
The Last Emporer
Empire of the Sun
Anything by Christopher Nolan (literally, all of them. Memento, Insomnia, Batman Begins, The Prestige, The Dark Knight, Inception)
Into the Wild
The Diving Bell and the Butterfly
The Godfather, Parts I and II (not III)
Big Night
The Usual Suspects (only if you don't know the ending)
Some of the better Charlie Kaufman films, particularly Being John Malkovich, Adaptation, and Eternal Sunshine of the Spotless Mind
Some of the Eastwood-directed films, particularly Unforgiven, Mystic River, Million Dollar Baby, and Letters from Iwo Jima
Michael Clayton
All of the Tom McCarthy films (The Station Agent, The Visitor, Win Win)
Leon
Some of the Paul Thomas Anderson films, particularly Boogie Nights and Magnolia, and There Will Be Blood if you don't mind watching a film exclusively for a single acting performance.
All Wes Anderson films, particularly Rushmore, The Royal Tenenbaums, and The Fantastic Mr. Fox
Leaving Las Vegas
Monster
The Hurt Locker
Gattaca
Almost anything by Danny Boyle, particularly Trainspotting, Slumdog Millionaire, and 127 Hours
Winter's Bone
Amelie
Run Lola Run
Heat
Amadeus
American Beauty
Good Night, and Good Luck
Donnie Darko
District 9
If you don't mind being super-depressed after watching a film, Requiem for a Dream
Black Swan
Shine
Gone Baby Gone
Das Boot
City of God
Children of Men
The Shawshank Redemption

If you're interested in Documentaries:
Grizzly Man
Touching the Void
Dark Days
Man on Wire
The King of Kong

That should get you off to a good start, let me know if you need more.