Today I installed PhpBB 3 forum on my linux hosting with MySQL database. Here are the quick steps:

1. download the package from www.phpbb.com/downloads
2. upload the contents of the package to the server
3. write-enable certain directories:
   chmod 777 cache chmod 777 files chmod 777 store chmod 777 images/avatars/upload chmod 666 config.php
4. open installation script in the browser:
   http://site.com/forum/install/
5. fill in the preferences and db connection details
6. delete the install directory
7. proceed to administration control panel (http://site.com/forum/adm/)
8. walk through all the settings, create categories and forums...

Links

PhpBB 3 Quick installation (original documentation)
PhpBB 3 users home