Generally the law that applies is the one in force in the place the copy is made. Obviously trying to enforce copyright on text documents included on Web sites would be laborious and irrelevant. By placing text and images on a Web site you are effectively giving visitors to your site the authority to make a copy on their desktop. This applies most commonly to software licenses. If you decide to make software available for download online ensure you have the license to distribute it internationally.
Confusion reigns over unmonitored chat rooms and message boards.
However, once someone tells them about illegal content they are obliged to remove it. Copyright law gets even more confusing when you start to consider caching, proxies and mirror sites, but it all comes back to the principle - who is doing what, when and where. What content is being downloaded, where is it being downloaded and who owns the copyright to it.