Well, I find myself tempted to do this against better judgment. My vague undefined criteria having something to do with wanting to find films that in some way represent as wide an array as I can manage of what I generally believe to be some of cinema's most important, interesting, or impressive trends, movements or national cinemas. A lot left unacknowledged of course, but let's see (trying not to second guess myself too much):
Faust - Murnau
An easy one. Probably the best surviving (as in the films) silent filmmaker, so he gets to speak for the era as a whole. Faust is an odd pick, since it's not his most famous, significant even roundly put-together works, but in my mind it exists as an astoundingly mad work of creative overflow, one of the best explorations of what's inherently seductive about pure visual storytelling. Could easily replace with The Last Laugh, though.
Touch of Evil - Welles
Another no-brainer, at least for Welles. I'm not his biggest admirer, but his breakthrough film redefined how artists and audiences think about cinematic language, and for me Touch of Evil represents something of his peak formal perfection. Not the colossus that Citizen Kane obviously is, but this film fucking rages.
The End of Summer - Ozu
My favorite Ozu film, so I'm indulging a bit. If a single artist has to stand in for the whole rich history of Japanese cinema is has to be him of Mizoguchi, and I don't quite think the latter has made such a profound wide-spread impression on every filmmaker who's since followed as Ozu. As for picking a specific film, I think his color work contains everything amazing his earlier work did, plus some of the most beautiful color theory in all of cinema, and The End of Summer is fucking perfect, a film about everything and a film executed with complete modesty and an unparalleled mastery of film language.
Hiroshima Mon Amour - Resnais
Duh, my only other true favorite to appear here probably. One of the landmark film of the French New Wave, needs little justification. As for the movement I'm tempted to go with Godard's Week End, as his role in this whole scheme speaks for itself and Week End, though a film I like considerably less than most of Resnais' work seems wholly important as a landmark film intent on fighting... itself, its audience, theology, philosophy, art, everything. Which makes it difficult to appreciate in a way let alone enjoy, but it's important and worth noting. But fuck it.
L'Eclisse - Antonioni
Not my favorite of his, as that would be La Notte, but any of his great trilogy here would be applicable. I'd go with this one since, though coming after L'Avventura the real breakthrough, L'Eclisse stands as his most daring and analytical formal triumph. Maybe the one film of his most interested in breaking apart the fundamentals of cinema and pushing it in various different directions at once.
Dog Star Man - Brakhage
True experimental cinema has to be represented one way or another, and while there are hundreds of artists that could easily take this place, few have the name-recognition or distinct clarity of artistic purpose of Stan Brakhage. So insert any one of the hundreds of brilliant films he's made, but I'll go again with one of his most famous (and feature-length).
Mirror - Tarkovsky
Another flag-bearer in a sense for his part of the world, I could have also gone with someone like Eisenstein who arguably left an even greater contribution to the development of film art. But Tarkovsky's Mirror stands out to me in terms of pure structure, visual elegance, and unfathomable personal communication (or is it exorcism?) of a director through his work.
Ali: Fear Eats the Soul - Fassbinder
It's Fassbinder, he's German and he's mindblowing. This is one of his richest films. So, uh... sure.
The Puppetmaster - Hou
Again, not my favorite Hou, but he's my favorite filmmaker and I really love almost all his work equally. This film or City of Sadness seems appropriate though in representing the New Taiwanese Cinema movement on the whole, along with the works of other masters such as Edward Yang, Tsai Ming-Liang or Wu Nien-Jen. The movement itself has a lot of significant formal, historical, cultural and ideological associations to it, which are too deep to get into in a post like this, but Hou's The Puppetmaster combines such innovative technique with the classic early tenants of the New Cinema's cinematic language and a startlingly honest if complicated portrayal of the country's rich history and culture through both incredibly personal storytelling and universal implications.
Close-Up - Kiarostami
Really, this feels to me like a fairly comparable film in terms of its formal accomplishment, relation to its own very important national cinema and culture, and overall significance to The Puppetmaster, only here for Iran. Beyond that, this is one of the absolute landmark works in examining the lines between fiction and documentary and the very nature of the filmic image.