Kamome has made an excellent technique for making amazing circles. This is a more detailed description:
Make a square selection
1. Open an image in Photoshop (screenshots are from Photoshop CS2 on Mac). Higher resolution and size will give better results. I have used a picture I took myself some time ago. If you want to use the same picture for this How to, you can find it here.
2. You need to make a selection that is squared. Use the selection tool, and set the aspect ratio to fixed:
3. Enter the same number in both fields. 1 and 1 works.
4. Make a selection.
Apply the filter
5. Apply the filter Polar Coordinates:
6. In this step, use the bottom radio button, marked “Polar to Rectangular”:
7. Now your image will turn into something that may look interesting at best, but we're not yet there:
Rotate and apply filter again
8. Rotate the selection 180 degrees. There are a few different ways to do this. One of them is to use the Edit menu, then Transform and Rotate 180 degrees.
9. Now apply the same filter we did in point 5. But do NOT just choose it from the top of the Filter menu. We are going to change one of the parameters so you have to choose it from Filters > Distort > Polar Coordinates… again.
This time, choose the upper radio button, labeled “Rectangular to Polar”:
Crop and save
10. Crop your picture to get rid of the stuff surrounding your new Amazing circle:
11. Voila!
Save and upload to Flickr.
12. I have created a group for Amazing circles. Please join!
You might want to have a look at all the other posts I have about Flickr!
Update 1: 100 members
Now 100 members on the Flickr group. Read about it here.
Update 2: 400 members
As of December 25, 2005, the Flickr group has over 400 members, and appproaching 3 500 amazing circles. Wow!
Update 3: Digged
Update 4: Amazing circles with GIMP
Flatfield at Flickr has posted instructions how to make Amazing Circles with GIMP.
Update 5: Dumpr.net
If you’re lazy 
just use the new automatic Amazing circles tool at Dumpr.net.