When the computer is off I plug in the network cable at the back of the pc (the cable links to an adsl modem). Funny thing is that the computer starts up automatically when I plug in the network cable. Why will that be?
It's because the setting is enabled that allows your PC to be brought out of Standby or Hibernation by another computer in your network. That could include a router. One workaround is to forget Standby and turn the machine off or you could post which Operating System you're using and someone will help you find that setting.