We need to know your iMac model. Please do "About this Mac" from your Apple menu and report what the resulting window displays for "Processor." The choices are G3, G4, G5, and Intel. Recommendations reqquire know which you have.
"About this Mac" will also display your version of the Mac operating system. You equipment profile now shows "iOS5." That is for iPhones, iPods, and iPads and cannot run on a Mac computer. If you update the OS info in your profile, you will get help faster because people don't have to ask for needed details.
If you have one of the older G3 iMac's with the CRT screen I would say that the resister that drops the screen voltage on shut down has failed this is why you are seeing a 'ghost' on the screen - abit like the very early days of old black and white televisions as when they were turned off you got the little bright white dot in the centre of the screen.