Using Adobe Photoshop Elements 8.0......?

Are you trying to erase everything outside of the selection? If your Layers palette shows you have only a Background layer, you need to promote it to a regular layer (easy as double-clicking). You can then make part of the image transparent either by erasing or by using Layer Masks (something like Layer > Layer Mask > Reveal Selection). You then have to save to a file format that allows transparency, such as PNG.

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