Let's check out some effective Magento Extensions and Magento Modules by Mage-world. All of our modules are developed in line with Magento standards and can be compatibly applied in all stores. They are of undeniably great use and quality with high rate of flexibility. More information: http://www.mage-world.com/magento-extensions.html
I personally prefer Simple Machines Forum (SMF) to phpBB, but there are a lot more forum applications available and a lot of things to consider depending on what role you want the forum to play on your site. It should be easy enough to integrate whatever forum you choose through Magento CMS.