I'm going to throw my hat in for The Thin Red Line because I've always found it to be a much better film than the one that got all the attention in 1998, aka Saving Private Ryan. It's an excellent film that, while much of the movie is either in the midst of combat or cowering behind the lines, it focuses so much on the individual lives of the soldiers who fought in Guadalcanal that it's much more of an introspective piece than a "real" war film. I also find it hard to disagree with the music or the emotional resonance.
That being said, as Wereduck point out, it's a long movie and it's pacing can be tedious. Characters are barely given names you can reasonably follow and their stories are told almost as if they were mere afterthoughts. It's definitely a movie you need to think through, and as many reviewers will point out it's more about the philosophy of war and the men who fight (hence the title) than the actual battle specifics. It also relies heavily on acting, flashbacks, and the viewer being invested in the story. I've recently re-watched it and it's still a fantastic movie.
As for others, many good ones have been pointed out so I won't reiterate them here. However, if you need a comedy, you can't go wrong with Dr. Strangelove. It may not be a traditional war film but it does deal with the Cold War in all of its wonderful black humour. Now that is certainly a classic.
(Has anyone mentioned documentaries? The War by Ken Burns is also worth a watch. Heavily American based but tells more personal stories like all of his other works.)
