...a photoshop image will always have straight sides, unless you position it in a layout application (indesign, quark or illustrator for example), inside a rounded clipping path or rounded frame...
...in photoshop, a background is either transparent, or colored, but regardless of that the images will always have cornered edges...
...first either hold down alt/option and double click the layer that says background, to make it a layer...
...or double click layer and click OK, same result...
...if you now select the white border with the magic wand tool and hit delete you will have a transparent background...
...however it is still technically still a square image with a transparent background...
Andrew