I Think phpBB is most widely used open source forum script. phpBB is written using PHP scripting language and uses MySQL database. It is very easy to install and has got a very good and easy to use Admin control panel. phpBB has got wide development community who can help you in solving any issues. You can easily create your own theme or there are lots of themes available that you can use free of cost. phpBB is the best open source forum script which can fulfill all your demands and my second choice Mybb.