This newbie would like to know how to backup his database?
Version 6.4.4
That would be done from the hosting side. You can do a database backup from within the admin panel but for the most part you would do it from cpanel or whatever you use on your hosting account. mysql would be a place to start.
As DrChaos said there is a backup utility in your admin panel. This utility depends on having access to the mysqldump binary.
phpMyAdmin is a nice tool for db management, and phpMyBackup (I think this is it) works well too.
If you have command line access:
/path-to-mysql-binary/mysqldump --opt --u <forumdbusername> --password <forumdbuserpassword> <forumdbname> > /path-to-backup-directory/forumdb.sql