If you are talking about recutting fabric strips for the pad with the rounded corners, then you probably mean you want to cut the new fabric on the bias for the binding.
The fold types can be confusing.
With quilts, almost everyone folds the fabric in half, sews the cut edges to the quilt, then turns the strip to sew it down on the other side. This creates *two* layers of fabric in the binding. Bindings are subjected to the most wear on a quilt, so having two layers of fabric in the binding helps the edge last longer. This is called "French fold" or "double fold" binding.
Often other types of sewing call for fabric that has two folds (cut edges towards the middle) for binding. This is what you see in bias tape. It is actually called "single fold" (have not looked this up to see why). Something like your changing pad with rounded corners (and potholders) often call for this type of binding because it is cut on the bias and has a *lot* of stretch in the middle -- which is where you need it when sewing rounded corners, at what will be the outside edge of the rounded corner. The folds make the edges more stable, but the single layer of bias fabric in the middle gets to stretch a *lot*. This is very helpful on, say, potholders where you need enormous stretch around tight rounded corners.
Connecting Threads has a series of tutorials on binding you might want to check out:
http://www.connectingthreads.com/tut...tutorials.html