Apply transparent background in GIMP

1. Select Layer → Transparency → Add Alpha Channel

2. Select the background using the Fuzzy Selection Tool (Magic Wand)

   and press the Del key.

GIF will allow you single-color transparency, while PNG has full alpha channel support.

1. Open the image and choose Select > By Color from the menu.
2. Click on the color you'd like to remove.
3. Select Colors > Color to Alpha from the menu and click OK.

This will result in the transparent, checkered background that you're looking for. If the background is made up of multiple colors, just repeat these three steps as necessary.