Raspberry Pi is a Linux machine, so it meant to be web accessible. One thing is to connect to internet from it and another is to run web server on it. Depending on what WWW features you want there are many options to do so. But probably most common way of serving websites is with PHP and MySQL. To get all of this working, there are several preparations required on Raspberry Pi. Drcurzon have shared his experience on instructable.
He thought that Raspberry Pi is low power device, so it can be turned on all time and serve as small webserver for non heavy loads. After initial RasPi setup he installed all component including Apache with PHP then MySQL and FTP for easy access and update web contents. phpMyAdmin for easy database managing. Each component requires user and password setups that are explained in instructable. If you follow all steps you should have your webserver up and running.