Yes, this function can be done easily with GIMP.
Don't know about GIMP yet? Learn about it
here.
First, create a new file (File > New)
Now create a new Layer (Layer > New)
In another window, copy-paste the image you are wanted to creat a trans-back for into this new file. Use the "select contiguous regions" tool (aka Magic Wand) to select the areas of the picture than you want to delete, and then press "Delete" on your keyboard. Continue to do this, until you are satisfied.
Now, select the "Layers" tab on the rightmost GIMP box. Click on the background layer (the one without your image), right click it, and scroll down to "Delete" and click it.
You should now see the image you want, and the part you want transparent should be a dark gray checkerbox.
Now go to Image > Mode and select "Indexed" Using the default settings should work fine.
Finally, save your image as a GIF (see
here to learn how to do that, if you don't already)
You should now have an image with a transparent background, yippie!