Partners is a new feature that allows your organization to become friends with another organization online via HootBoard. Once you partner up you can "push" posts as an admin from your board to theirs. For example, maybe you have an upcoming event that would be great for the members of their community to attend as well. Once partnered, you can send that same post to your partner's board so they find out about the event too.
Here's a video that covers everything you need to know about Partners.