Most image editing software should allow you to look at and edit the histogram. You'll probably find the histogram shifted to the right, i.e. there's a gap between the lowest data (sky background) and what is regarded as black (value of 0 by default).
You should then be able to slide the "black point" up from zero to redefine what this black point is. Slide it just up to the histogram and your background should be much blacker.
If you later apply some noise reduction (e.g. in Photoshop) you'll find a gap reappear as the lowest values in the image are largely noise. Simply slide it a bit up again after noise reduction and it'll look even better.
The benefits of being able to redefine black with a camera means we can shoot images even in light pollution - wish I could do the same with my eyeball so visual astronomy in the suburbs was possible as well!
Paul