One of my favourites, assuming you don't want to code your own, is phpBB click here
Another excellent choice is Invision click here
It very much depends on what you want out of the bulletin board and your ISP limitations on its use within your account. Some ISPs allow one MySQL database but will not allow it for discussion boards. Others allow multiple MySQL databases and don't seem to mind what use you put them to.
Other considerations can include attachments in relation to bandwidth use. If you allow people to post images or send files to one another (or even if you use the board to send out multiple emailshots to members) it can all eat into your available bandwidth very quickly.
phpBB is freeware to most users, but you should check out the licensing terms to make sure your intended use fits into its conditions.
As far as server demands go, obviously the more traffic your board gets the greater the demand on your web server. This is something you may want to ask your web host about since they should be able to advise you based on your account type and their existing experience of available boards that other users may have on their sites.
Just don't go istalling a board because you can. I rarely use discussion boards on sites and even when I do I they are normally closed off for specific corporate use rather than general public access. Identify the need and/or probably traffic before you go ahead.