The creator/owner of a website is not automatically responsible for any and all content that happens to be posted on said site, but this does not mean they are absolved from all responsibility either. Just as the owner of a physical business property is not automatically responsible for everything that happens on their property, still, they must take steps to prove that due diligence was taken to prevent illegal acts, or to inform authorities when such acts took place.
The most common form of website liability is copyright infringement, such as a person posting video or audio that is owned by a third party. But they also must be aware of defamation of character, threats, and obscenity or child pornography.
Cyberspace is a unique entity in a legal sense, however, and proving how much an owner knew about illegal activity can be very difficult. But to be legally "covered", a system of internal controls and checks needs to be demonstrated by the owner, as well as posted disclaimers and warnings and user agreements about user posts or other use of the site that may be deemed illegal. Any site that has commercial traffic or message boards usually has legal consultation before going live for just such situations. That is, if they are smart.