There are two types of group:
public groups and their contents can be seen by all members of the community and everyone is free to choose to join the group by clicking on Join the group,
private groups and their contents can only be seen by members who you have attached to the group.
You can create groups by clicking on the button Publish in the community in your profile page and then on the Public group or Private group link depending on circumstances.
Then assign a title to your group, describe its purpose and add an image. Register: your group has been created. It will appear from now on in the Groups tab in your profile as well as in the Groups page.
Click on the name of the group to access the page listing its contents. You can then add an article or some media.
In this page, use the button Manage the group to add people directly (and possibly allow them to be group administrators like you).
Caution, if you have created a private group, you must without fail use this function to add participants.
Use the People link to manage the different members of the group, change their status and possibly exclude them from the group.
All existing groups are presented on the Groups page; private groups are marked by a padlock icon in front of their names.