I've tasted on a number of occasions coffee that has smokey flavor. Is this a natural flavor of the beans or was it acquired in other means (e.g. during the roasting process)?
It's all about the roasting process. The beans themselves have a much subtler flavor, a bit closer to tea really.
Try tasting a light roast coffee compared to a medium roast or dark roast at your local coffee house. The differences are readily apparent. The coffee you describe was most likely a dark roast which generally has a quite smoky and even burnt character.
Any kind of added flavor notes take place during the roasting process. Have you ever had a beer with added hops, or a fruitier beer made with orange peel? Almost the same effect.
You can take a traditional bean and do all sorts of fun stuff with it to give the taste some character and let it stand out.
There's so many varieties out there you can really create just about anything. Whether or not it'll taste good is another story. Ginger coffee comes to mind....gross.