Over the years, I’ve used many website forums and bulletin boards. They’ve been based on a variety of packages, such as phpBB, LiveCloud, or vBulletin. PhpBB is, I think, the reigning king of forum software.
To add a forum to a site, you can use an off-the-shelf system, or roll your own. Casual sleuthing will reveal an assortment of at least 50 some-odd off-the-shelf bulletin board systems, from which you can choose one to run your forum. If you want a bespoke forum, most frameworks provide bits of the necessary functionality, if not entire customizable forum modules. For example, see the Django forum applications or Drupal’s Forum module.
Web forums are OK, but I’ve yet to encounter one that wasn’t clumsy in some way — sometimes, in multiple ways. And you know what? I’ve yet to find one that beats august Usenet, when accessed with a decent reader such as Agent or Unison.