clipped from: wvs.topleftpixel.com   
king circle || canon350d/efs10-22@10 | 1/250s | f8 | ISO200 | handheld

1- First you need a 360 panorama image. You can use partial non-360 panoramic images too but a full 360 will look more natural. For that you can use commercial software like PTGui or Photoshop's Photomerge, or use freeware applications like AutoStitch.
1a- note that you can use a wide landscape shot instead of a panorama too, but it's best if the width is at least twice the height of the image, and also make sure the left side and right side of the photo match more or less. To check that you can try using Filter>Other>Offset command in Photoshop.


2- Open the image in Photoshop and resize it to a 1:1 square image. To do that you need to uncheck Constrain Proportions in Image Size panel. I recommend scaling up the smaller size to match the bigger length and not the other way around. So for example if your photo is 2000x1000 pixels resize it to 2000x2000 pixels.


3- Rotate the image 180 degrees.


4- Go to Filter>Distort>Polar Coordinates and select Rectangular to Polar.