A Trojan or Trojan Horse is a small program or piece of code which is loaded on to your PC pretending to be something innocent like a small program, music or video file, that's why the P2P programs like Kazaa, iMesh and the like are so dangerous. When this Trojan is on your PC it can send info about you, your passwords and other information you would like to keep secret, to a third party. It can also allow that third party to gain access to your PC. It is named after the Wooden Horse which was supposed to be a gift from the Greeks to the Trojans, but in fact was a way for the Greeks to enter Troy hidden in the belly of the wooden horse (if my memory of that day in school is correct).
If the shortcut is appearing on your taskbar, then check your Startup folder and also msconfig and make sure there's nothing in there, to do with this shortcut, starting at bootup. To get to msconfig, go to Start > Run and type msconfig, click OK and go to the Startup tab, untick any mention of the shortcut from there. If you can't get rid of the desktop shortcut by right clicking on it and choosing Delete, then try to delete it from Safe Mode.