I know this is an old thread, but I wanted to point out something.
If you just want to create the crescent shape, no extra modding or transparency, then you don't even need to do anything besides layer the white ellipse over the black the way it was when you started.
Using Subtract on the Pathfinder panel will give you the same effect, except the top layer will become transparent no matter what color your fill and stroke are and you can see through the black circle where they overlapped. However, the white/top ellipse still exists and you can edit your ellipse by moving the subtracted layer around because it is really just a compound path.
If you want to create the crescent as a shape itself you can use any of the first three tools on the bottom row of the pathfinder pallet: Divide, Trim and Merge. You can see what I mean if you look at the layers in the screenshots below. (I used the Divide tool for this example.)
When you apply any of these with both shapes selected, you end up with cutout areas where shapes overlap: the bottom layer(s) are cut out. Then you can delete the shapes you don't want leaving you with your new shape. You don't need to worry about layers or paths and you can add a stroke to it like any normal shape.
I prefer to use Divide because it offers more options to play around with. It basically divides your shapes along every intersecting line... think of it like turning your shapes into a jigsaw puzzle where all the pieces fit together exactly.