Select the area that you want to delete with the wand tool, and go to Edit > Clear.
EDIT: Oh, and you have to have an alpha channel added, if you don't already. Right click on the layer you're working on and select "Add Alpha Channel".
An even easier way would just to select the fill tool, choose the color white, and in one of the drop down menus, click color-erase that way it erases the white background for you.
You have an image with a white background, but don't want to go through and erase all of the pixels manually.
