There are many free forum scripts available, all pretty much the same
Most common are phpbb and smf. These you install on your MySQL database and your hosting account.
However if you dont have the knowhow or are on a free host, there are certain site which will give you a free forum. There are hundreds so search google for them.
SMF, invision free, phpbbfree, the list just goes on, the only drawback is most are ad supported, which lets face it no one likes.
Also with a remote hosted forum you do not have access to the raw files and can only have the templates/mods that the free host has put on for you.
Your best off getting some hosting and doing the forum script yourself, off hand i know phpbb have a support forum that has plenty of staff "in the know" to fix any problems you may have.
Good luck, Tkey