The background is important and the image itself is in png format. I'm using photoshop but can probably get access to other applications or even shareware.
This task can be done using the "Marquee tool" option of Photoshop. Open the image in Photoshop, select the background using the "Marquee tool" and simply delete it. This will remove the background and make it transparent.
Open the image in Photoshop. Use whatever tool you like (magic wand, lasso, marquee) to select the area you'd like to make transparent.
Make sure the image is not on the background layer. You can tell by looking in the layers palette, a background layer will be called, surprisingly enough, "Background" and its name will be italicized. If you need to, you can convert from a background layer to a standard layer by double-clicking the layer in layers palette. A "New Layer" dialog will open, asking what you want the layer to be called (Photoshop will suggest something imaginative like "Layer 0"). Pick a name and press "OK".
You can now hot "Delete" to make the selected area transparent.
Of course, once you place the resulting image into a layout program, you don't get a "transparent" background, you get a "white" background. I'm not exactly sure what effect you are aiming for, so it's hard to know what to advise you, but I would usually create a "clipping path" for something like this, so that when the final photoshop image is placed into your layout program (eg Quark Xpress or Adobe Indesign) there will be just the image itself and no background. The image can then be placed over a colored background or over another image. Use the Photoshop help and look up "clipping paths". Another way is to create a "mask" in Photoshop, but that is a little more advanced.
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