Join us on May 14-15, 2019 at Forrester Digital Transformation & Innovation 2019 in Chicago, USA.
In short, use departments when you are trying to represent a user's placement within an organizational chart, and use groups to represent the various functions a user might carry out.
To manage groups, access go to Setup Users > Setup Groups. Note that each group created can be tied to a specific organization, although not necessarily so. As mentioned earlier, a user can belong to multiple groups and, unless restricted, a group can have members that spans span several organizations.