QUOTE
his morning I went to wake up the computer, and SuperDuper was on the screen, but it had not run.
From that and other statements you made, I'm going to assume you are running the iMac for long periods of. Maybe even 24/7 as I do. With that assumption I will hazard the following: It may just be a combination heat and badly seated RAM chip(s). Also, check System Profiler to see how much RAM and in which slots/banks the System sees. Even a "About this Mac..." should show that you have 4GB.
It's quite easy to remove and reset the RAM chip(s) in that model. Turn the iMac off, of course. On your other computer, <watch the video at this site!> This is a 5+ minute medium resolution video, but there is a low one there, also.
Assuming that doesn't fix the problem and you have two RAM chips, try removing the one on the left and restart. If things work OK, the chip you removed may be bad. If it didn't help, remove the one that's in there with the one you took out. If you still have problems, either both chips are bad or there is something else wrong.
If the chip re-seating/swapping did not help but you can still run, try to download a temperature monitoring app. There are several around, I use <mcFanControl>s, but others may offer other choices. These apps will show you temps at various places in the iMac. If you find any higher than ~175°F you might want to check the cooling fan speeds. At that temp probably all of them should be running well above their idle speed.
Let us know what you find. And check back for other suggestions, too.