For a forum, I'd recommend you get at least 1 GB Diskspace and 10 GB Bandwidth, but each forum is different. You can get free hosting if you are not willing to or cannot afford paid hosting.
The amount of people who use topics such as this to advertise their web hosts is quite appalling.
Anyway, bandwidth is simply the amount of data transferred from your web server to your visitors "browser". Each page load will consume bandwidth. If you happen to use up all your bandwidth your site will be inaccessible till the following month when your bandwidth is reset. Of course a simple Google search would have told you what bandwidth is.
As well as simply visitors using bandwidth, scripts that pull data from external sources, as well as accessing the site not via port 80 will contribute to "bandwidth" (depending on how your host or ISP calculates).
To add on to what I said before, softwares such as vbulletin and ipb have options to load most of their data from google engines instead of using all of your resources you have for your site, thus making it faster and more for your visitors. 😉