Target edges for sharpening: See the demo for "Sharp" (the functionality in "edge" is the same as that offered via the checkbox in "Sharp"). The benefit of using using the "Edge" button for this work is that you can create multiple manual edge selections for a highly customized result as needed.
Target surfaces for noise reduction: This is the inverse of the previous scenario. You may apply a surface mask to a noise reduction layer in order to help avoid using noise reduction on important edge details. Generally noise reduction algoryms do a relatively better job of this, so this often is not as impactful as masking the sharpening layer.
Create black and white line art:
Refine existing edges: This allows you to try to automatically fix bad edges or cut out subjects.