For questions about the Organic Groups module, not the Group module. It enables users to create and manage groups.
The Organic groups module is a contributed module that enables users to create and manage their own groups. Each group can have subscribers, and maintains a group home page where subscribers communicate among themselves.
They do so by posting the usual node types: blog, story, page. A block is shown on the group home page which facilitates these posts. The block also provides summary information about the group.
Groups may be selective or not. Selective groups require approval in order to become a member, or even invitation-only groups. There are lots of preferences to configure groups as you need.