A darning foot is used when you drop the feed dogs and are doing free-motion quilting. It 'hops' as you stitch which allows you to move the fabric freely in any direction.
A walking foot is used when you leave the feed dogs up. It helps pull the top layer of fabric through at the same rate as the bottom layer, helping to prevent pleats and puckers when working on several layers such as you have in a quilt sandwich. It is not absolutely necessary, but you will probably end up with a nicer finished product if you use one. I even use my walking foot when attaching the binding, it comes out smoother and straighter that way.