There are no circular images. There are only rectangular images. But there can be images where corners are transparent, so that only a circle shows.
In light of this:
First, make sure that your layer has an “alpha channel”.
Create a circular selection with the “Ellipse select tool” (the 2nd one in the toolbox). Use the “Tool options” dialog
If the selection isn’t perfect on the first try, you can move it (click around the middle) or extend it (click inside, near a border or a corner).
Once you have the required selection, invert the selection (Select → Invert, or Ctrl-I) so that everything is selected, except your circle.
Erase the selection (Edit → Clear or [Delete] key). You should have your central circle left, surrounded by a checkerboard pattern. (this checkerboard is not part of the image, it just indicates the transparent parts of the image).
You can reduce the checkerboard to the minimum by auto-cropping the image (Image → Autocrop image)
Last, save the image in a format that supports transparency, like PNG (JPEG doesn’t support transparent images…)
If you are going to work further on the picture, save it as XCF (Gimp native format).
GIMP Tutorial - Making a Circle-Shaped Image by Ofnuts is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.