If Chinatown perfected the noir genre, to me The Third Man defined it. Carol Reed's classic is what most people think of when the word "noir" comes to mind. Black and white, set during war time with it as the background, a trench-coat wearing main protagonist who has a somewhat mysterious past, and absolutely gorgeous shadows and lighting.
Oh, and this movie had the added benefit of one Orson Welles. Yeah.
The main theme for the movie, played on a zither, is as famous as it is strange. It's a good tune, don't get me wrong, it's just...strange.
If you want to start digging your hands into the noir genre, this is as good as a start as anyone can get (I haven't watched The Maltese Falcon so don't quote me on that). A definite recommendation.