Kostya

Where to begin where to begin.

I say start with Kurosawa, the man (if he is indeed a man and not some kind of God) is incomparable.

His masterpieces:

Ikuru, Rashomon, Seven Samurai, Yojimbo, Hidden Fortress, Throne of Blood and (I believe) the apogee of half a century of filmmaking - Ran.

Silent film hasn't been mentioned, and is a dead art these days, but I highly reccomend you watch some of the silent masters, their techniques are still relevent today.

F. W. Marnau - Sunrise, The Last Laugh and the first vampire movie ever: Nosferatu the Vampire. He also did a production of Goethe's Faust but I don't know how good it is.

Sergei Eisenstien - Battleship Potemkin also his 1939 film Alexander Nevsky is remarkable in its originality, it's also a film which Stalin personally interfered with, a fascinating collision of propaganda and art.

Fritz Lang - Metropolis - See also M

In terms of comedy I'm not getting into the Keaton vs Chaplin debate but certainly The General, Sherlock Junior, The Great Dictator and City Lights are classics of the genre.

Moving on.

I believe the 30's horror films contain some of the most original and striking technical aspects of any era. James Whale's productions especially.

Frankenstein and Bride of Frankenstein are must sees.

The Mummy.

Phantom of the Opera

Browning's Dracula is also very good.

I'm also putting in the original King Kong, for its revolutionary special effects, and status as one of the first films to show particularly gruesome deaths onscreen.

Ingmar Bergman's been mentioned: Seventh Seal is his masterpiece.

Orson Welles, what an incomparable fellow. Citizen Kane is obviously considered his masterpiece, but as Rodney has already said A Touch of Evil is equally brilliant.

Hitchcock, the master of suspense, somebody has already listed his stuff.

However, Clouzot's Wages of Fear is one of the earliest, (and in my opinion best) examples of suspense.

Fellini's 8'1/2 is a must see.

Visconti's The Leopard

Nobody quite renders eroticism and lust like Bertolucci's Last Tango in Paris

Noir:

The Big Sleep, Maltese Falcon, Treasure of Sierra Madre and of course Casablanca immerse yourself in Bogart and marvel at a dead genre.

Then go and find a copy of Polanski's Chinatown, and Ridley Scott's neo noir masterpiece Blade Runner (that's right Blade Runner is noir) and finally take a look at the Coen Brother's homage The Man Who Wasn't There

Werner Herzog's twin masterpieces:

Nosferatu The Vampire (remake of Marnau's original)
and the unforgettable

Aguirre: Wrath of God

Westerns.

Tracing back to the early days:

High Noon and The Searchers are both essential to understanding the genre and the stylistic artifice which developed.

Then you must move to (in my humble opinion) the king of all western directors (I know the Howard Hawkes and John Ford camps are going to come after me for this) Sergio Leone.

Now, you might think Spaghetti Westerns are a waste of time, but I say to you that they are one of the most visually striking and stylistically intricate genres of all time.

Absolutely must see:

A Fist Full of Dollars - A reworking of Kurosawa's Yojimbo, interesting to compare the two.

For a Few Dollars More.

The Good the Bad and the Ugly.

Once Upon a Time in the West - Best Western ever made.

Then you gotta look into the revisionists.

Peckinpah led with The Wild Bunch but not long after that Eastwood directed High Plains Drifter a bizarre reexamination of the Western through the lens of borderline cultural nihilism.

For a long time nothing of interest happened in the Western until finally.

Eastwood directed Unforgiven, his best movie, and probably the final chapter in the genre, a brilliant film.

Horror returned in the 50's with Romero and Ed Wood taking a whole new tack on the genre.

Ed Wood's films, every single one of them, is a lesson on how not to make a film.

Plan Nine From Outer Space is considered to be the worst film ever by many, I disagree, but in any case it's highly entertaining.

Night of the Living Dead represents a watershed in horror, and indeed film history, the first film to have the evil guys win at the end (that I can think of)

Kubrick's The Shining and The Omen created a new kind of horror in the 70's.

Speaking of Kubrick, the man's entire catalogue is brilliant. Clockwork Orange, Barry Lyndon, Lolita, 2001: A Space Odyssey and of course Dr. Strangelove are all quite frankly masterful pieces of film.

The 70's also saw the arrival of a peculiar breed of films spearheaded and championed by a young and still brilliant Martin Scorsese, starting with Mean Streets, moving on to Raging Bull and culminating in the timeless classic Taxi Driver which all dealt with themes of loneliness and alienation, urban forms and social decay. Add to this Francis Ford Coppola's timely meditation upon technology and invasion of personal space and privacy, responsibility and paranoia - The Conversation

Gangster, a brilliant and somewhat underrated genre, one which places human affairs into a framework of life and death.

You must go back to the original Scarface from 1935.

Then basically, 1971 - The Godfather and The Godfather Part II gave rise to the modern gangster film.

High points since then: Scarface, Goodfellas and Donnie Brasco

The highly contentious, and very long (4 hours 20 minutes) Once Upon a Time in America is probably worth a look if you have the time, some say it is a masterpiece, others say it is a load of shite.
You must also check out Elia Kazan's On the Waterfront to comprehend Brando's brilliance, and to a lesser extent Viva Zapata.

War:

Holy crap what a gigantic genre (if it can be called a genre.)

Um

The old ensemble WW2 romps aren't masterpieces, but oh so enjoyable: Where Eagles Dare, The Great Escape and the Dirty Dozen to name a few.

Varmalov's Stalingrad is probably a must see.

Kurosawa's Ran which I have already reccomended, but I put it here to reinforce it's brilliance, is a war film and more, watch it like RIGHT NOW!

Kubrick's Dr. Strangelove, I already mentioned this somewhere too I think the best ever exploration of the realities and absurdities of the post nuclear age committed to film.

Then there's the Nam movies.

Apocalypse Now Coppola's masterpiece, see Redux if you can. It's not really a war movie as such, but a film that transcends genre and style which happens to be set in a war.

The Deer Hunter Cimino's masterpiece make's me cry every time.

Platoon Stone goes for realism and in my opinion fails to soar as high as Cimino or Coppola, but it remains a touchstone for the genre.

Kung Fu.

Holy crap, there's so many classics of the genre, but I'll attempt to give you the lowdown on my personal picks:

Drunken Master II: The Legend of Drunken Master - Jackie Chan's best movie by far. The choreography is phenomenal, this was made in the early 90's and makes The Matrix look hackneyed.

Enter the Dragon: Bruce Lee's only English language movie, it's also his most accessible and a classic of the genre. Lee's presence in this movie makes it all the more worthwhile, not to mention the meat and potatoes Southern Chinese style of Kung Fu, which unlike the Wushu practices by say Jet Li, is not about performance, and all about kicking ass.

The Golden Harvest Bruce Lee movies: The Big Boss (aka Fists of Fury) and The Chinese Connection (aka Fist of Fury) rad. Also, if you can get a hold of the uncut version of Game of Death yuo can watch the extended version of Bruce kicking the crap out of Abu Kamal, awesome. And of course, finally Way of the Dragon anyone who wants to see Chuck Norris get his ass kicked should watch this movie...

Once Upon a Time in China and Fist of Legend are Jet Li's best 90's films. Also look at Shaolin Temple his first major film, absolutely brilliant.

The Shaw Brothers - Five Deadly Venoms and Invincible Shaolin.

The Eight Diagram Pole Fighter, The Seven Grandmasters, 36th Chamber of the Shaolin, Iron Monkey (1993 version), Incredible Kung Fu Mission.

The list goes on. Not a genre that is stylisitically or technically brilliant, but the choreography and direction is unique and dynamic as hell. Also, let's face it, it's freaking rad.
Miscellaneous:

I'm abandoning my genre divisions as they are causing me too much grief, but bear with me.

Um,

Laurence of Arabia - I mean come on, if you haven't seen it, what the hell, but chances are you haven't watched it with a critical eye, and it's an entirely different experience.

Crimes and Misdemeanours Woody Allen takes morality and basically scares the shit out of you, while making you laugh...

Jack Nicolson movies:

Easy Rider - Holy shit, one of the most stark, basic and effective indictments of the impossibility of American ideals in America, and an ending which is unapologetic in its utter emptiness. Cult classic

One Flew Over the Cuckoos Nest - Acting and direction in this film cannot be described.

Patton - George C Scott's greatest ever performance, and a brilliant and honest look at war and the people who fight them.

The Hustler - Paul Newman rules as one of the first, (if not the first) anti-heroes, Fast Eddie Felson. George C. Scott's ruthless performance as Bob Gordon and the overwhelming grittiness and dark themes for a film made in 1960 make this a must see. Best pool movie ever.

A Man for All Seasons - Schofield rules and this movie is truly one of the best adaptations of theatre I've seen.

The French Connection - Popeye Doyle and friends descend into the underworld and beyond, cop drama at its best.

Cool Hand Luke - Newman again, best chain gang movie ever.

Ok, well it's midnight, I've probably forgotten about a million movie I should have put in here and tomorrow I'll be frantically adding them piecemeal throughout the day, but I guess this should tide you over for a whiles...

I kind of like movies a bit...

Just a tad...