The icon, or the file, you are working with might be assigned to another user on the system, so the Ownership or Permissions are not granted to you to delete them. To see these, Right click or Ctrl + Click if you have a one button mouse on the file in question, choose "Get Info" from the drop down menu. It should open another window and one of the sections is "Ownership and Permissions". Change the permissions to the file to your name, might need to enter a password, and then try to delete the file. Picture of window with the ownership and permissions area visable.