There are a lot of free forum hosting and there are website that will help you set up a forum quickly and easily for a certain amount of money which you pay monthly for the hosting cost, but there are also free applications and servers you can use from the open-source community that you can do yourself. For example, MySQL database service, Apache web service + PHP, and phpBB
phpBB.com down 
(website down but downloads are linked). You might also need to setup your own SMTP service for registration but that's optional. Also registering for a domain name.