blender: why can't I difference modify this?

Check the file attached. Why can't I do a Boolean modify: difference to cut the little shape out of the big shape?

Because the normals on your outer shape are not correct. Select the outer shape, go to Edit, select all vertices and then on the menu to the left, Mesh Tools, click on Normals: Recalculate.

If you are trying to hollow the object you will have to move the inner object down a bit or at least its lower vertices other wise the Boolean will just create a void in the larger shape.

For hollowing another approach is to delete all the bottom faces of the shape and then use the Solidify Modifier.

Good luck

Hmm neither of those methods worked for me. Here is another simpler model I'm working on. I want to punch a cylinder half way~ into the triangle.

Once again your normals are wrong on the prism.

I figured as much. However, employing the technique of selecting all verts in edit mode and clicking recalculate normals did not fix it.

PLus the geometry of your prism is odd. You seem to have overlapping faces, duplicate vertices and unnecessary diagonals. The prism only needs to be 3 rectangles and 2 triangles.

Remove doubles (vertices).
Remove the diagonal edges.
CHeck the normals.
Try again, should work