Common misconception, actually. The proper term is that it's a message board divided into forums divided further into sub-forums. However, some people skip the message board part and just refer to the entire message board entity as the "forums" because when they look at the message board that's what they see: a list of forums.
That sounds awful specific, have you got a source? Wikipedia says the terms message board and forum are interchangeable.
Internet
forum, or forums, or message board are interchangeable. However, even the wikipedia article goes on to keep saying "forum
s" when referring to the entire entity, rather than just forum.
Here, you can look at vBulletin's official forum:
http://www.vbulletin.com/forum/forumdisplay.php/5-Site-FeedbackYou can see that the general name up top for the forum is just "Forum." However, each individual forum, i.e. Site Feedback is called a Forum. Then, there is another forum within the Site Feedback forum called "vBulletin Jira Questions and Quick Tips" that is explicitly called a Sub-forum.
Same as with another top message board software developer, phpBB:
http://www.phpbb.com/community/index.php the entire thing is called Forums, but then there are Forums, further divided into Sub-forums which are Forums within Forums.
The people who made the technology for this board, SMF, are a bit different:
http://www.simplemachines.org/community/index.php?P=3c70dec8a136446af0b17b29700f94e2& they call each area a "Board," with what I call Subforums being "Child boards."
Really, there isn't one standard definition, but Forum/s > Forum > Subforum is the most common. Forum > Subforum > Subsubforum/Child subforum is rare, though.