An activity is permission for a user to carry out a particular role or piece of functionality. This is normally the functionality related to a in individual plug-in, and generally offers read-only and read-write options.
Owning the necessary activity is not sufficient for a user to run a specific piece of functionality - they must also have the appropriate plug-ins installed on their computer.
Activities are irrelevant on single-user versions of Epic (such as the demo) as the user automatically has all activities.