The background is not a layer. It is the background. There is a subtle difference.
Make the image into a layer by double clicking the Background in the Layers palette. Then it will be made into a layer and you'll be able to make parts of it transparent, to where you can see the checkerboard (indicating transparency).
A transparent layer only allows you to "see through" to the layer(s) or background underneath. Make the background and/or lower layers invisible (click the eyeball icon) or delete them to be able to see the checkerboard pattern through the transparency of the layer above.
If you're trying to make a frame you can paste over other images, select the photo and delete the image out of the inside. Delete all other layers in the image, and save as Frame.psd or something. Then later you can open that image, copy the layer, and paste it over other images.
Hope this helps!
First make sure you are using a layer. This means the name of the layer in the layer palette should not say Background. If is does double click it and press Enter to convert it to a layer. Once when you are on a layer to make it transparent use one of the following:
select area with any selection tool and press Delete
use the Eraser tool
drag the transparancy slider in the layer's palette
use a layer mask - you'll find a button at the bottom of the layers palette.
Does the file have layers? or is it a single layer titled "Background"?
If it has multiple layers over a solid color background layer, just delete the background layer and use the marquee tool to delete the frame.
If it is a background layer, then duplicate the background layer, making it a new layer. Delete the old background layer. Then proceed to marquee and delete frame.
When done either save the file as TIFF, GIF or PNG and you should have a transparent background. Don't flatten the file though, this will automatically use your foreground color as the background solid fill color or will fill in the transparent areas for you.
Hope this was helpful.