Share with groups for boards
Sharing a board with a group makes it available/visible to that group.
Share board to group
To share a board to a group click on
Select the groups you want to share the board to and click OK.
Why do I see a star/asterisk(*) next to my groups?
This means that there is a difference between the published (live) board versus your current revision of the board. You may have added a new group or deleted a group from the board. Once you've publish the asterisk will disappear.
Make sure to click PUBLISH / PUBLISH CHANGES to make the change live.