I’ll mention another war film:
The Bridge on the River Kwai
Here’s a classic for the list: Dark Victory. It was on TCM a couple of nights ago and I was just blown away by Bette Davis (although I think Humphrey Bogart was woefully miscast as the Irish horse trainer).
How about
Musicals
Seven Brides for Seven Brothers
Brigadoon
An American in Paris
Funny Face (or almost any Fred Astaire movie)
Biographies
Sgt. York
Pride of the Yankees
The Benny Goodman Story
The Glenn Miller Story
The Buddy Holly Story
Comedies
My Man Godfrey (either version, but Carole Lombard’s is my favorite)
Father Goose
Some Like it Hot
Bringing Up Baby
StG
I’ll just second His Girl Friday. Love it!
Maureen, our tastes in movies is so alike! I weep at the very thought of An Affair to Remember.
I tend to vomit at the first mention of An Affair To Remember.
“That’s Entertainment”
Clips from all the old MGM musicals. For our younger readers, this film is like the Cliff notes version of American Film Musicals 101.
Plus, just like MTV, you only need the attention span of a fruit fly to enjoy it.
Local Hero (Burt Lancaster)
The Private Life of Sherlock Holmes (Robert Stephens)
Last of the Dogmen (Tom Berenger)
Along Came Jones (Loretta Young)
Harvey (James Stewart)
Greenfingers (Clive Owen)
Okay, there are a ton more that I could list, so these are just a start. Perhaps not all of these are critical successes, but enjoyable nonetheless.
Harold and Maude
Alice’s Restaurant
King of Hearts (short)
Leaving Nornal
Easy Rider
The Third Man
Hmmm - close call. It depends on which day you ask me. Some days I’ll say Red River; some days I’ll say My Darling Clementine. So let’s add Clementine to the list.
Since we’re throwing in Cary Grant films, don’t forget Arsenic and Old Lace - shows what a great comic actor Grant was.
And the old Ealing comedies with Alec Guiness: Kind Hearts and Coronets, The Lavender Hill Mob, andThe Lady Killers.
Hmm… I’ve seen a whole lot more of the movies on that page than ones in this thread, but I think they’re right about the " you may not know" part since I don’t know anyone who’s seen House of Games, which is a pretty good movie, even if it’s not as gripping as The Grifters. And they put one movie on there twice 
I’ve seen House of Games! My SO and I are total Mamet dorks. We do Mamet-inspired skits all the time.
SO: Do you understand?
Me: Yes.
SO: Do you?
Me: Yes. Yes.
And so forth. Is Glengarry Glen Ross on this list? I need to at least pretend to contribute.
ZJ
Dr. Strangelove; or How i learned to stop worrying and love the bomb
(yes, that’s the entire title 
)
Cool Hand Luke.
Nobody can eat 50 eggs.
Hey, a guy who is arrested for cutting the heads off of parking meters can eat as many eggs as he wants.
Silent era movies for whippersnappers:
Sunrise (1927).
The Black Pirate (1926). Whaa? Big budget color movies in the 1920s?
Seven Chances (1925). Buster Keaton, and just damn funny.
Man With a Movie Camera (1929). As eye popping as any music video today.
Earth (1930).