You will need to sign up for a web host (like Godaddy.com or Aplus.net). For a small forum a shared hosting plan is fine. Try to find one that includes the option to install a forum for free (Godaddy does this).
If your hosting plan doesn't include forum installation, you can easily it yourself. You will need to create a mysql database. Then download your forum software (phpbb.com is a good free one). Follow the installation instructions.