I didn't watched many korean movies, so my suggestions are limited:
In addition to the ones suggested, if you like rom-coms you can watch White Valentine, Il Mare, Windstruck, A Man Who Was Superman, all with Jeon Ji-hyun.
Barking Dogs Never Bites has dark humour, Memories of Murder and Mother are excellent crime thrillers. The Host is also a good movie, all by the same director.
You can watch Kim ki-duk ones for some artistic and gritting movies but also moving (3-iron, "Spring Summer, Autumn, Winter ... and Spring", Bad Guy, Samaria, Dream, Time, or The Island - and much more);
There's the raunchy Sex Is Zero 1 (and 2), kind of korean American Pie, first one has Ha Ji-won. 200 Pounds Beauty is also famous (and the actress is hot).
A depressive one is Christmas in August.
Watch JSA, Swiri, Silmido and Taegukgi, Friend/Chingu for thriller/crime/drama.
Avoid D-War.
Good movies from what I know, I have yet to watch: The King and the Clown, The Good, The Bad and The Weird, A Bittersweet Life, Poetry, Green Fish, Peppermint Candy, Secret Sunshine.
Edit: I also enjoyed Hahaha, it's a good comedy, clever and fun but without being silly.
Edit 2: Also Attack of the Gas Station, has Kim Soroo, Have to watch it some day.
For more, not necessarily best:
http://en.wikipedia.org/wiki/List_of_highest-grossing_films_in_South_Korea