The Group module allows for creating arbitrary collections of your content and users on your site, and grant access control permissions on those collections. It is available as of D7, and has a D8 version also. It is designed to be an alternative to Organic Groups.
Organic Groups allows content itself to be groups, which isn't always what people want. It relies on an entity reference field to keep track of the ties between a group (node, term, ...) and its content (node, term, user, ...)
Group instead creates groups as entities, making them fully fieldable, extensible and exportable. Every group can have users, roles and permissions attached to it (refer to "What are the various roles supported by the Group module?" for more details about that). Groups can also act as a parent of any type of entity. Group provides an extensive API to attach entities to groups.
Advantages of Group as compared to OG
- Group started in D7 only, taking full advantage of entities introduced in D7.
- Group doesn't 'abuse' nodes by adding custom fields to them in order to make the module work.
- Group does not allow you to have per-group roles or permissions. These are instead defined on the group type. So group types, group roles and permissions are configuration instead of content. Because of this approach, group types and roles exportable (with or without Features).
- Organic Groups is commonly perceived as a huge module, which carries a lot of "legacy" from pre-D7 releases and makes it tougher to ever get upgraded to D8.
- A D8 version of Group is already available (not so for Organic Groups).
Disadvantages of Group as compared to OG
- As of today, Group only has alfa / beta releases (not any official release).
- Even though some say it is self documenting, I think it is not (maybe because I don't want to digest the source code to find out what this module can do ...).
- Not a lot of documentation available.
- As of today, there is no group tag on drupal.SE, though there is an organic-groups tag, and there are already quite a few Group specific questions. Anybody interested in creating it (again), feel free to reuse my previously approved tag wiki and/or tag excerpt (no need for credits ...).
Integration with various modules
The Group module integrates with various other (popular) modules, including:
The beauty, IMO, of these integrations are that they are more or less out-of-the box (not dozens of extra modules needed to make those integrations work).
Refer to Integrations with other modules and its "Related issues" about other modules for which integrations already exist, or are in the pipeline.
What people say about the Group module
Similar to what you see in the Group issue queue: most (if not all) people who know about Group just love it ...