Allowing new posts
This option allows you to restrict students from posting new content in this forum.
For most forums you will want to leave students unrestricted and choose the first option to allow them to start new discussion topics (threads), and also to post replies within those threads.
Sometimes, however, you will want to disable this ability. For example, this is useful for the News forum when you only want teachers to post new items that appear on the course main page. In this case you might choose the third option "No discussions, no replies".
Sometimes you might want to only allow teachers to start new discussions, but still allow students to reply within those threads (for example within the news forum on the site home page). In this case you would choose the second option, "No discussions, but replies are allowed".