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. If you would like to restrict access of a board to a specific course, you can add the board to a Bundle and it will only be visible to students who own that bundle.
Your students cannot create boards or groups, they can only create a post in a board that you've made.