Select the image area you want to have visible. You can do this by first selecting the background and clicking "Select>Inverse." It'll give you a good grab.
Next, copy and paste the selection to a new RGB file. Now turn off the background layer.
If you have PS 5.5 or higher version, click "File>Export to Web."
This feature will give you the option of a transparent background. There's a check box to click for activating that.
Those are the steps I take. Could be different for lots of folks.