How do I remove an icon from the Desktop. When I drag it into the trash it reappears.

A little more info might help. What is the icon for and what put it on the desktop? Is a new one being recreated with one in the trash remaining, or is it not staying in the trash?

Have you tried rebooting?

Trash – Can’t Empty or Move File To Trash

Is it one of the items covered by Finder -> Preferences

And if you have a .dmg mounted, it can be one of the above.