1. phpBB
There are tons of forums online that operate on phpBB. It’s free and even comes with a variety of design choices, so your forum doesn’t have to look exactly the same as all the others online. You can see phpBB in action here and here.
2. Simple-Press
Simple-Press is a WordPress plugin, so it’s extremely easy to set it up to work with your blog. It’s not as feature-rich as other forum tools, but it gets the job done. You can see it in action here and here.
3. bbPress
bbPress was created by Automattic, the company behind WordPress, so it makes sense that it integrates seamlessly with WordPress blogs and sites. It doesn’t offer all of the functionality that you’ll find in forum tools like vBulletin (see below), but it gets the job done for simple forums. You can see it in action here and here.
And if you’re willing to spend some money on your forum, then I’d recommend checking out vBulletin, which you can see in action here or here.