By far the greatest place for cinephiles in Seoul is the Cinematheque, located near exit 5 of the subway sub Chongno-3(sam)-ga, which is located on lines number 1, 3 and 5, although access is quicker from line 5. It is not the easiest place to find, but if you keep straight after coming up the stairs of exit 5 and go under the overpass you should see film posters on your left. Go into the building and up to the 4th floor. You should see the box office outside.
The cinematheque has a website, and although it is in Korean, you can navigate through well enough in English: http://www.cinematheque.seoul.kr/
The schedule of films is really quite strong. There have recently been retrospectives on Raoul Walsh, Fritz Lang, Stanley Kubrick, and Jean-Pierre Melville. Currently there is a Friends of the Cinematheque festival with a wide variety of items, including mini-retros on Francois Truffaut and Abel Ferrara, plus a showing of Rivette's CELINE AND JULIE GO BOATING, Scorsese' TAXI DRIVER, and many others.
I have yet to attend a Film Forum screening, but it is located at the same place as the cinematheque. It focuses on contemporary films outside of the mainstream, basically the modern day art cinema of North America. The website is here: http://www.filmforum.co.kr/, although it is less user friendly than the cinematheque.