It'll does sound permissions-related, but repairing permissions via eg Disk Utility won't sort it out - that only deals with system files, and your desktop is stored in your user folder.
If you Get Info on your home folder (highlight it in Finder and hit Cmd + I), down the bottom is a Sharing & Permissions section. You need to make sure your account is listed there - it should specifically say "(Me)" after it - and that the privilege level next to it is Read & Write. The other users / groups listed don't matter so much, but should be set to Read Only.
Assuming that all checks out, there's a button at the very bottom of the panel that looks like a cog, which should have an option to "Apply to enclosed items". That takes a while to complete (a few minutes, usually less than a dozen), but odds are you'll be sorted afterwards. Maybe reboot a couple of times to be sure.
You may need to click the little padlock in the lower-right and enter your password before you can make changes.