First, if you're running Snow Leopard, you ave an Intel-based iMac, and not a PPC based G5 iSight iMac. The late 2006 iMacs look similar to the iSight iMacs.
Here are some items to try:
Run Disk Utility to Repair the Hard Drive and Repair Disk Permissions
Place your original, came with the iMac when purchased install disk, into the optical drive slot on the iMac. Restart the computer. Immediately press and hold the C key. (If you have upgraded your OS from when you purchased your iMac, example your iMac came with OS 10.3 installed, and you're now using OS 10.5, then use the OS 10.5 Retail Install disk that you had used to upgrade your OS instead of the original, came with the iMac disk.)
Release the C key when the spinning gear below the dark gray Apple logo appears. Wait for the installer to finish loading. DO NOT do an OS installation. After the installer loads select your language and click on the Continue button. When the menu bar appears at the top screen, click on Utilities, and pull down to select Disk Utility. On the left side of the window that opens, select your normal hard drive entry (manufacturer's ID and drive size). In the lower portion of the Disk Utility window you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say Verified, then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.)
If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid at the top middle of the window if it isn't already selected. Now click the Repair Disk button from the lower right area of the window. If the disk utility reports any errors that have been fixed, then re-run Repair Disk again, until no errors are reported. When that is done, click the Repair Permissions button to the left of the Repair Disk button you clicked earlier.
When that is done, quit Disk Utility.
If Disk Utility reports errors it cannot fix, then you will need DiskWarrior and/or Tech Tool Pro to repair the drive. These are third-party disk repair utilities that cost money. If you don't have them, Google them for more info. If neither of them can fix the drive, you will need to backup your data ASAP.
If that doesn't work, try resetting the System Management Controller (SMC)
1) Shutdown the computer.
2) Unplug all cables from the back of the computer, including the power cable.
3) Wait fifteen seconds.
4) Attach the computer's power cord.
5) Wait five seconds, then press the power button to turn on the computer.
Reset the PRAM or NVRAM
Reset your PRAM. Press and hold down the Command Option P R keys while starting your computer. You will hear the startup chime. continue holding down those keys until you hear the startup chime a second time. Release the keys. If the computer restarts, you will need to reset your Date and Time. It might be time to replace your PRAM battery.
Run the Apple Hardware Test (AHT)
See here for running AHT for Intel-based iMacs: