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
1:1
.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.