There are one or two forums that only use text files as the underlying database but they still require PHP support. I would always recommend that you have or arrange web hosting with full PHP and MySQL support. This allows you a 'proper' database which offers far more potential flexibility.
One of my own favourites is phpBB click here
It is an excellent product and well worth looking into.
If you have simpler needs and your web host supports FrontPage Server Extensions you can use FrontPage to generate a discussion board in a few mouse clicks. It is primitive by comparison to phpBB but it works beautifully and is drop-dead easy to get up and running with.
Another good option is YABB (Yet Another Bulletin Board) and if you don't mind paying, Invision Power Board click here is about the best of the bunch.
Another thing you need to be aware of is that unless you spend the time and effort moderating the forum your guests could post all sorts of things on there, some of which could potentially get you into hot water.
There is a lot more to it than just bolting a bulleting board into your site and you should make sure you know what you are getting yourself into before even tihnking about trying it out.