Here are a few ways to make a transparent background with LunaPic.
The best and easiest way is the Transparency Tool to remove the background.
However, you may want something else
Here is a tutorial on adding a Sunset to the sky
You can also use the Paint Bucket Tool to fill in areas with a color or pattern.
Another way, is to cutout an item and replace it on a new background. Use the Scissors Tool the Edit -> Paste Image
Finally, you should check out