A logo but that's most likely a given other than that,
1. A user friendly interface and mobile optimization (I can't tell you the amount of times I've tried to go on a forum with my phone and it was just pure chaos and frustration).
2. A search functionality. Sometimes I want to be able to see if someone has already posted about something similar, so I don't accidentally create a thread that already exists but the forum doesn't have search functionality.
3. Notifications and Subscriptions. Some members love getting notified when a thread or post they've created gets a reply so it is important to have notifications and allow members to subscribe to posts.
4. PMing. It is incredibly useful and it's a great feature when you need to speak to someone privately.
5. Spam & Security protection. Too many times I see a forum that has been littered with spam and it all could have been prevented if they had CAPCHA or email verifications. Also having moderation tools for members to report/flag inappropriate content.
There is a lot of features a forum owner should implement on their forums but this is just the top 5 that I think are crucial for a successful forum.