You can embed forums on your website with ease using Vanilla Forums. Once you've requested your forum from here and you have admin access, login to your forum. On the Top left you'll see a menu item that says Dashboard, go through the following:
- Banner: Change Forum Name
- Outgoing Email: Add correct name & email
- Categories: Create forum Categories, you can drag & drop to have parent/child categories as well.
- <embed> Forum: Copy that script onto the page you want to display the forum on your website.
To embed a category by adding this scipt before the embed script from your forum:
<script>window.location.hash = "/categories/forum-category-name"</script>
 Connecting user registration with your website
With jsConnect you can allows users register on your site to register on the forums with the same username. If your website is on WordPress use this plugin and if your website is on Drupal use this module. More detailed instructions here.
Vanilla Forums Documentation