War Film, Psychological Thriller, Epic Quest
It's no big revelation to call Apocalypse Now a war film. You obviously didn't think this was a rom-com. But what kind of war film is it?
Apocalypse Now definitely seems to be more in the anti-war camp, along with All Quiet on the Western Front, The Grand Illusion, and Paths of Glory. In fact, Coppola has said that he believes that all movies are by necessity anti-war movies.
Most films portray war as meaningless slaughter, not as a great struggle for ideals. That might be because they deal with more controversial wars: Vietnam in Apocalypse Now and World War I in those other films we just mentioned. Movies about World War II still portray war as brutal and terrifying, but they usually suggest that the struggle against Hitler was the right thing to do and that the sacrifice was worth it. Same for films about the American Revolution.
Apocalypse Now is a psychological thriller because—well, just look at Kurtz. Willard's a troubled soul, too, and he shares his psychological journey in his voiceover narration. The movie takes us into a world where war has messed with the minds of both the hero and the villain.
Finally, the movie's an epic quest because it depicts a sweeping historical event on a large scale (the Vietnam War) and features a main character going on a long-distance journey to confront a mysterious and frightening enemy. The film, like the novel on which it's based, can also be considered a powerful social commentary on the nature of "civilization" and the illusion of "progress."
The moral of this story is that it's all a jungle out there.