A messageboard group is a separator for a separate section of messageboards.
For example, take a look at this screenshot:
In the screenshot above there are 4 messageboards. The top three are not in any group, but the bottom one is in the "Marketing" group. For every group made, there will be a separate row on the discuss page with all the messageboards in that group.
You only need groups if you make a large number of boards and want them to be better organized. Currently you cannot restrict access to a board for a specific course, but this option will be coming soon and you do have the ability to create multiple boards for each course. Your students cannot create boards or groups, they can only create a post in a board that you've made.