It really depends on the kind of forum you want to host as in software. There are many free PHP forum software packages available (phpBB, Invision Board) to name a few. As well, as the ever popular vBulletin (not free). Each of them require at least 1 mySQL database and should run on the majority of PHP installs that hosts would have these days.
Depending on your free host, you may even get an auto-installer included for phpBB. Good luck.