Groups has been a feature of Microsoft's Windows Live service before it was integrated into OneDrive. Each group received 1 Gigabyte of storage that did not affect user storage quotas and was shared between all group members.
All group members were allowed to read, create, modify and delete contents -- meaning files and folders -- with a group's folder. Groups provided members with additional features, for instance group discussions, a calendar, options to mass email group members using outlook.com, or options to to upload and share photos easily.
The OneDrive integration of Windows Live's Group feature did not change that initially, but if you check out the groups page right now, you will notice that it is no longer possible to create new groups on OneDrive.
The page states that you cannot create new groups or invite people to existing groups in OneDrive. Groups that have been created previous to that are still accessible by all group members though.
According to Microsoft support members, it should still be possible to invite other users to groups even though the Groups page itself states otherwise.
The support members recommend that users use the sharing functionality that OneDrive offers instead. While that enables you to share files with others, it does not offer the same functionality and approach as groups yet.
While you can share folders with users so that they can edit files inside, you do not get the same manageability that groups provided.
While users that you have shared a folder or file with are displayed on OneDrive as well, the current feature lacks discussions, calendars and other management features at the time of writing.
This means that you will need to use another service to make use of those if you want to replicate the groups functionality using the Shared feature on OneDrive.
There is for instance no option to quickly mail all group members. Another difference is that you share using your own storage quota and not the group's fixed quota that is independent from a user's quota.
Microsoft has not released a statement yet why it removed the creation of new groups from OneDrive. If you are looking for an alternative, try out Yahoo Groups which offers a similar but not identical feature set.
Have you used Windows Live Groups or OneDrive's Groups feature? If so, what is your take on the change?