What defines the transparent parts and opaque parts is the alpha channel, which is an editable black-and-white overlay on the image-- in gimp the easiest way to do this is to edit the layer mask.
(the other way is to bring up the "channels" dock, but the selecting and deselecting of channels is annoying in gimp.)
You edit a layer mask by:
-starting with your image
-right-clicking the layer in the layers tab/dock thing
-Add layer mask, Transfer layer's alpha channel
Now in your layers box you see two thumbnails for the layer instead of one: the left one is the image in RGB, the right one is the B/W alpha.


Simply toggle these option on the right-click menu:
-Show Layer Mask makes the mask visible, if you want to look at it in B+W (I recommend you do)
-Edit Layer Mask, whether you're working on the transparency mask or not.
-Disable Layer Mask disables transparency, and if you also hide the mask then you can see the true color of the transparent pixels.
-Before you save Apply Layer Mask (you can ctrl+z this after you save to unapply it)
Basically when editing the mask, black means see-through. You can copy/paste, edit the contrast, paint, whatever on the layer mask.