Making a web forum requires a server side technology, with a database. It's even more complicated than a CMS, there are features such as profiles, feeds, knowledge base etc that regular CMS don't even have out of the box.
So if you want a forum, you'd better use one that's already developed and make it your own. That's what 99,99% of people who want a forum do, because it's a very hard and long task to create, develop, debug and maintain one.
For years and years, the king of forum engines has been PHPBB. It required PHP on the server as well as a MySQL database. It's old tech and comes with too much possibilities I think, I wouldn't recommend it in 2015.
The forum we're on here, Webflows' forum, is built on a much newer program: Discourse. www.discourse.org. It's free I guess you can grab it and have someone install it on your server.
I hope this helps.