Japanese movies and tv series tend to be more powerful than movies made in other countries. I don't know why but they appear to be more realistic in their depiction of life and so their fictional characters appear more real than from movies made in other countries, which makes their movies more moving.
I think that's because in the Western world fictional narratives focus on the hero's journey (something that is not routine) and not on the more common aspects of life, while Japanese fiction, including even the craziest fantasy animated series, tend to develop a much stronger domestic and more realistic characters side than western fictional works. Western fiction portrays characters that are much more extrovert and much less risk averse than real people (they are heroes and not ordinary folk) while Japanese fictional characters are more introvert and risk averse, thus more like real people (they are just ordinary folk).
Some of the stuff I think are a must:
These were among the movies I though deserved a
.
Kurosawa
-- Rashomon

-- Ikiru

-- Seven Samurai

-- Red Beard

-- Dersu Uzala (Soviet film, though)

-- Ran

-- Dreams
He is Japan and Asia's greatest director in history. If one wants to be literate about films one must watch at least a dozen Kurosawa films.
Ozu
-- Late Spring

-- Tokyo Story

-- An Autumn Afternoon
Ozu is perhaps the most humanistic of directors (at least from those which I know something about). No other film director managed to touch the human soul so naturally and with as much depth as Ozu. As Kurosawa once described Ozu's films: like still water with strong currents underneath.
Though I still haven't watched a large proportion of his films, so I can only give three masterpieces to the table. I think that the average quality of his films is superior to Kurosawa's, even though he is slightly less influential.
Mizoguchi
-- Ugetsu

-- Sansho Dayu
Mizoguchi is another great director. Though I must admit that his filmmaking skills were significantly lower than Ozu's and Kurosawa's. His movies fell less "natural" on average (sorry for my lack of my English vocabulary) and pacing is a bit off, though he was the most popular Japanese director among Western critics in the 1950's and 1960's, thanks to his exotic period dramas, and despite the relative difficulty I found in consuming his work I think that movies like Ugetsu are still incredibly powerful and unforgettable experiences.
Miyazaki
-- Future Boy Conan (TV series)
-- Nausicaa
-- Castle in the Sky
-- My Neighbor Totoro
-- Kiki's Delivery Service
-- Porco Rosso
-- Princess Mononoke
-- Spirited Away
-- Howl's Moving Castle
The greatest Japanese living director, greatest animator in history and also my favorite director overall.
Takahata
-- Grave of the Fireflies
-- Only Yesterday
Another animation director, not as good as Miyazaki on average but when he gets things right his movies are as moving as Ozu's and more adapted to the modern zeitgeist.
Mainly due to the rise of the manga, since the 1980's the animation industry became much bigger than live action cinema in Japan. Even Kurosawa complained in the late 1980's that animation had drawn out all the talent from film.
Most animation in Japan is made for TV and video, Ghibli is the only animation studio out of the over 430 animation studios in the association of animation studios of Japan that focuses exclusively on film, the others produce mostly TV and direct to video products. As a result I think that watching stuff like the anime series Neon Genesis Evangelion is a must, even though it is extremely silly it is also a extremely powerful and unforgettable series. Even more silly but perhaps even more powerful and quite disturbing is another anime series I am watching lately, Madoka, it is about magical girls and it's genre that it's fits in are drama, horror and tragedy, it's target demography are young adult males.