There are 2 options. 1. Use a joomla forum component, like "Kunena". 2. Use an external forum like phpbb3 or vbulleting and find a bridge like jfusion to integrate it with joomla If you go with 1 then installation and integration will be smooth and easy but you will have less features. And if you go with 2 you will have a powerful forum intergrated in Joomla with amazing features but believe me you will pass thru many issues, bugs and integration difficulties. If you have no good knowledge in php/html/css I suggest so go with option 1.
You can always use vBulletin in your joomla if you want you just need to integrate it using a bridge.
Work out what features you need exactly and then compare with the ones mentioned already. I use Kunena and it is suitable for my purposes.
kunena is the best forum component for Joomla. I have used it in one of my website and it have all the facilities and it automatically integrate with Joomla users.
kunena is the best at the moment.You may try ninjaboard also.It is an emerging forum component and have many features.
you can use kunena, but it has no more options, I used it before and changed because of that. Kunena is still new and lack of options. Use phpBB with joomla
Kunena is the best if you want free component. But if paid is ok for you, then I suggest you to choose agora. It is very similar to vbulletin and costs only 30$