Would somebody be a d'ear for me?

I've never used paint.NET before, but the idea is the same regardless of editor: you either remove what you don't want or cut&paste what you want to a clean, transparent background. Transparent backgrounds are displayed as gray checkerboard in almost all (if not all) editors.

Using paint.NET on your original request:

 Open the image in paint.NET.

[spoiler=In the toolbar to the left, click on the Magic Wand. This "smart-selects" areas of consistent color or shape. Click on the white background around the object. You may need to raise or lower the Tolerance of the Magic Wand at the top.][/spoiler]

 Hit Delete to delete the white area.

 Clean it up.

You could zoom in and delete all the remaining border whitespace. Instead...

To me, the blue ring looks like it's supposed to fade into the background. To create this fade, select the object (I used circular select, but you can hand-select around it).

 Spoiler

Go to Edit > Invert Selection. This inverts the selection and thus selects the empty border area.

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Go to Effects > Blurs > Gaussian Blur. This blurs the selection, creating a "fade". Play with the Radius setting until you get a satisfactory effect (I used a radius of 5).

 Spoiler

 Images of it cleaned up and not (on black backgrounds).

The fade could look better, but apparently feather-select in paint.NET is a plugin I'd have to download to use. Feather "softens" the edges of a selection so there would be a sort of gradient between the blue ring and the blurred area.

Save the image as .gif or .png. JPEG does not support transparency. If you're doing this for a GIF animation, you'll have to remove the whitespace from each frame and save it as an animated GIF (versus a flattened one).