Open or create an image. Pick white as background color, and select it with the Magic wand tool. Once the selection made, delete it (Delete key).

You get an image with a transparent background (grey & white squares as a default option).

When you have a transparent background, go to: File> Export> Gif89a Export. A window will appear, as shown below. Choose your options. Click OK and save your image.

This is the same image. It appears better on a light background because it was created on a light background from the start. The Magic wand tool has a hard time selecting colors with a non rectangular image. This is the reason you have to pay attention to the color your going to delete. The problem is due to anti-aliasing, used to shape borders, and the resulting "pixelized" effect.