For a new project, I wanted to create a shadow on an icon I was creating in order to give it some depth. I couldn't find any good visual tutorials on how to do this in GIMP, which usually means I'm not using the right keywords, but I was able to find a forum post: Creating an Elliptical Vignette in GIMP.
Following the posters steps exactly didn't get me what I wanted, but modifying his steps slightly, I was able to do what I wanted, so I figured I'm write down what I deduced from his process and include some screen shots in an effort to help anyone else who wanted to do something like this.
Step 1: Prep Your Image
I prepared my icon for a shadow by increasing the cancas space a bit so that there was room for the shadow below the icon itself. The result was as follows:
Step 2: Ellipse Select
Next I selected an area beneith the image where my shadow was going to go.
Step 3: Feather the Selection
Next we're going to feather our selection. This is what will give us the gradiant. To feather your selection, you have to use your menu, and go to Select -> Feather. Gimp will prompt you for how many pixels you want to feather, I went with 25. Your visual for your elliptical selection will go away when you feather it, but don't worry, it's still there.
Step 4: Fill with FG Color
Last, you'll want to fill with the color you want to use as your shadow. I just went with black, and set black as my foreground color. From there, we go again to the menu, and go to Edit -> Fill with FG Color. You can repeat the fill if you want your shadow to grow darker. I had to do a fill 3 times before I got the following result:
Hope this helps!