[the 'stache]

Some other favorites:

On the Waterfront-Kazan and Brando. Nothing more need be said.
Rififi-the twenty + minute heist is one of the greatest scenes in cinematic history.
Cat on a Hot Tin Roof-Newman, Taylor and Ives are as formidable a threesome as you'll see in the 50s.
Brooklyn-Saoirse Ronan is a revelation, a fantastic actress with a bright future; the story is simple, yet gripping.
Persona-another masterwork from Ingmar Bergman, and one of my personal favorites.
Libeled Lady-by this point in her career, Jean Harlow's comedic sense is virtually unparalleled (save for Carole Lombard). A wonderful screwball comedy with a great cast, and Harlow is absolutely magnetic.
Double Indemnity-if somebody held a gun to my head, and asked me to pick a favorite noir film, I'd have a hard time not picking this gem. MacMurray and Robinson are great, but this is Barbara Stanwyck's film. She is the quintessence of the femme fatale. Maybe the greatest performance of, perhaps, Hollywood's most versatile actresses.
Tokyo Story-the film that served as my introduction to Yasujirô Ozu. Some sixty years after it was shot, Ozu's wonderful drama about the post WWII family dynamic still casts a huge shadow. A must see for every film enthusiast. Just beautifully done.

And, my favorite two films of the new millennium...

Mulholland Drive-David Lynch's magnum opus. A veritable textbook on how non-linear timelines can be handled in film. See it once, and you'll think about it for weeks. Naomi Watts and Laura Harring are wonderful together. Silencio! No hay banda!
Lost in Translation-Sofia Coppola hits every note perfectly. Part love story, part homage to Tokyo. I never get tired of this film, no matter how many times I watch it. Bill Murray and Scarlett Johansson have great chemistry together.