Just curious: what is the point of pinnin and unpinning a board from the header menu? If Does this help me move boards between groups/ organizations? That would be a great feature, but it doesn't seem to allow me to do this. Thanks!
Unpinning a board from your header menu means that the board will not show up in the menu that is displayed when you click on the "Boards" button in the top right corner of the screen. For people who have many boards, it is often useful to unpin boards that are less frequently access to minimize the number of items in the boards menu.
Likewise, pinning a board causes it to be listed in that same menu. This can be useful if you want to keep up with a public or organization board that you are not a member of.
You can switch which organization a board is related to, but it has nothing to do with the pin/unpin feature. To change the organization that a board is associated with, click the "Board Menu" button in the righthand sidebar just beneath the "Add Members" button. In the menu that pops up, click "Profile". Within the Board Profile screen you can change the organization. See the image below.