Kunena is one of the most popular Joomla forum components. It is very easy to install and use. Kunena is developed specifically for Joomla and takes advantage of all its functionality. Additionally Kunena adds a constantly increasing set of features that will help you manage your communication with ease.
Agora is a fully featured forum component for Joomla. It includes various styles making it possible to easily match the design of any website. It also includes advanced group permissions, attachment managerment, ban and warning systems. Agora is also fully functional with Community Builder and one of the most common choices for Joomla webmasters.
The ccBoard is a Joomla 1.5 Native simple forum. The most important provided functionalities are moderations, community features, ranks, avatar, CAPTCHA, CB/jomsocial integeration, Bad words filter, SEO/SEF, IP logging Latest Post Module and Popular Posts Module. This makes it a very good addition to your Joomla website.