Start off from the restore disk or a snow leopard retail disk (whatever you have), insert DVD, restart and keep the C key pressed until it boots up - slowly off the DVD.
Wait until the restore or install screen appears, do not re-install but disk utility will be an option in the Utilities section of the menu bar.
Run Disk utility and repair the disk (run a couple of times) while your mac is booted from the DVD and not the hard drive.
the files are not corrupted, its the disk structure (software keeping track of files on the disk)
attempt a proper repair first, if that succeeds, make sure you have a back-up in place. If you can run a time machine back-up now then give it a try, but dont delay
Do as louishen has advised running Repair Disk from the install DVD and see what is reported. If still problems you have a failing hard drive. Seagate and WD warrant their drives for three years.
Yes icons moving about can be precisely this problem.
It is highly unlikely. But, anytime you make a major modification to your disk or operating system, it's always best to have a backup. Better safe than sorry.
If you have data you treasure on that hard drive, you should be in the habit of making backups. Hard disks, after all, are mechanical devices. And as we all know, eventually any device with moving parts will fail. So having a backup is always prudent. And with the price of external hard drives being very affordable, there's little excuse not to have Time Machine backing up your data regularly.
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