The Facebook Like Box is really designed to use the URL for a Facebook page, not for a general Web Page. It pulls Timeline content from a fan page within Facebook.
It can be used for a Web Page if the page has defined Open Graph Meta Tags and has been Liked at least once. However, there will be no "stream" functionality. The Like-Box for a Web Page would just have functionality similar to the Facepile plugin (except that Facepile just shows you friends, and a Like-Box also shows non-friends.)
Typically, though, you would want to supply the Like-Box with a Facebook Page URL in the following format:
http://www.facebook.com/pages/Facebook-Page-Name/Facebook-Page-ID
Perhaps you are intending to use a Like Button or Facepile Social Plugin instead? Either of those can accept any URL as input.