I think you may be confusing the terms "border" and "binding".
Borders go on before quilting and are quilted with the quilt. Borders are like the frame of the quilt. Some quilts have many borders some have none.
Binding goes on after quilting to cover the raw edge. Bindings are generally only 1/4 to 1/2 inch wide. All quilts (except rag quilts) will have some kind of a binding.
I hope this points you in the right direction. Ask more questions if you are still confused.