Hi,
A few questions for you:
1)What printer are you using?
2) What RIP program?
3) What program are you using to create or set up the art?
Now some answers!
Generally, the printer is going to print whatever you send to it. So if you make the background black, it's going to print black ink around your image. If you make the background white, it will print white ink around your image.
What you need is a transparent background. That means getting rid of the black background layer. I would recommend saving your file as a Photoshop PSD or a PNG file in order to preserve the transparent background.
Then, when you send your artwork to the printer, just the image will print and not the background.
-Alex