In many ways, groups are Action Network's fundamental organizing unit. Each group has its own email list. Subscribing to one list does not subscribe you to another, and similarly unsubscribing from one group does not unsubscribe you from another.
Groups sponsor actions, like petitions, and anyone who takes that action is subscribed to the group's list. Activists are de-duplicated by email address within a group's list, but not across groups. When you write email, you target that email to one and only one group.
All this means that typically each organization would have one group on Action Network, which holds their email list. To subdivide and target various segments within that list, people typically would use tags. However, in some situations where you have totally separate email streams and subscribe/unsubscribe list membership, you'd want separate groups.
Beyond that, groups accomplish a few functions. First, they allow organizations to have a branded presence on the site and publish actions under that brand. Your organization can create a group, upload your logo, and get your own group email list, so all activists who take action on pages published under your group are added to that group's separate email list.
Second, groups allow organizers to work together in more formal ways. Anyone can't just join a group -- organizers apply and are approved or are invited by group administrators, and you can have multiple administrators for your group, each with their own permissions.
Organizers in your group can publish actions under your group's brand, but they cannot access to group's email list -- only administrators get that access. So, when organizers create an action, they'll have the option to make that action sponsored by your group. Activists who take action will then be added to the organizer's personal email list because they created the action (unless they've removed themselves as a creator) and the group's email list at the same time. This sets up a useful federated structure -- you can imagine local field staff or volunteers being organizers in your group and creating their own local petitions, events, and forms under the group's brand to help win the local campaigns they're working on. Over time, these local organizers are building up their personal email lists, so they can easily keep in touch with their local activists. But all of those activists are also added to your group's national email list, which you as an administrator have access to so you can keep in touch with those activists about national campaigns.
Administrators in your group can have individual permissions controlling whether they can send emails, download group data, or edit group settings and invite or approve new organizers. This allows many people to work together effectively administering your group and its email list and database.
Groups also come with a discussion board so group organizers and admins can plan campaigns in private. And groups can apply for access to two additional action types -- campaign pages and event campaigns.