I'm talking about backings that have two pieces of fabric running in one direction and another running perpendicular. Thusly:
Wide backings are a popular choice, but keep in mind that they are often not made from the same high quality greige goods your regular quilt shop fabric is made from. So, wide backings can also be hard to square and they can be squirrely.
I also understand that longarmers aren't fans of using bedsheets as backings. this is because the thread count on sheets is often too high to easily quilt through. However, though I haven't used a sheet as a back before, once I scored a used duvet cover from Crate and Barrel for under ten bucks and used that as a backing and it was just fine. But, it wasn't really a sheet, and because it was Crate and Barrel, it was of reasonably high quality.
This is the super soft, furry stuff that is so wonderful to touch. Longarmers don't seem to mind minky fabric on the back of quilts. Remember that quilting stitches will really disappear into your quilt if you use minky fabric on the back, so if you're wanting to highlight the quilting on the back, minky might not be your best choice.
A note about using minky: please don't use the yucky stuff they have at the discount fabric stores. That stuff feels okay (not as good as quilt shop Cuddle) but once you wash it, yuck. And contrary to popular belief, it isn't too bad to quilt with, as long as you use basting spray and not pins to baste.
Quilting at home
So all of the things above that I mentioned that longarmers aren't fans of? Well, they're often more do-able at home. If you're basting your quilt on a table or the floor, you don't have gravity working against you, and you're always going to be attaching the backing to the batting. No sag! The other thing about quilting from home is that you can deal more easily with things like pieced backs.
Ah yes, quilting at home. That will be a whole new series!