It is a weird but common problem. Go to Firefox's Preferences and change the download folder. Change it to the Documents folder in your hard drive, either in your user home directory or the Documents folder at the root of your hard drive. Download a file. If that works you can change it back to your Desktop. Do not log in as root! The problem will reappear when you log back into your regular account.
If that doesn't clear it up, then you have a bad permissions problem with your Desktop.