How do I outline (determine/specify) a polygonal area, and then cut it out (eliminate it) so the lower layer shows through?
It's pretty simple. The second part of the solution is the same however you create the selection, whether it be a shape selected with the Fuzzy-Selection tool, using the Paths tool, or using the Free-Select tool.
The simplest method to do is to use the Free-Select tool:
and then create the selection.
You can either draw the lines by hand by pressing and holding the left mouse button, or by drawing the lines by clicking to start and then again to finish that line segment. If you use the second method you can also press Ctrl and have the lines snap to angles:
Then, once you complete the selection, all you have to do is make sure the correct layer is selected...
... and press Del, giving you this result:
I hope this helped!