You have 2 different routes to use a forum on your Joomla! website.
The first is to use a a forum like phpbb and other 3rd party forum systems. In this case, the website and the forum run independently from one another. This means 2 user accounts, and users having to log in twice when switching from the website to the forum.
A solution here is to use a bridge which connects the two together. Allowing for users to only have to create one account and sign in once. The bridge handles the rest.
The other route you can go, is use a forum that was made for joomla! specifically, and will run within joomla! itself. An example of that forum would be fireboard. Fireboard is installed like anything else with Joomla! by going to "Extensions" and then "Install/Uninstall" and doing what you usually do.
The pro and cons? The 3rd party forums sometimes can have a lot more features, where as the ones made for joomla!, what they have is what you got, and there's no way around it(unless you know how to program).
Either way, this is how you do it. If you want a no brainier install. Get fireboard. Its got a fairly good amount of features that is only growing. If you don't mind taking the time for a bridge, etc, get phpbb3.
If you need help dong the actual install, just let me know.