Yes you will need to upload to the 'public_html' folder.
If you will only be running a forum on your site, you could upload the
contents of the 'upload' folder to directly to the public_html folder.
If you want to have your forum on a subpage of your website (ie.:
www.mysite.com/forum) you should upload the 'upload'
directory to the public_html folder. You should rename the 'upload' folder, either before or after FTP, to the name you want to use (if you want
www.mysite.com/forum, then rename the upload folder to 'forum').