Dan,
For the first iteration of Groups available to churches, just think of it as a replacement of People Lists. However, it does lay a foundation for future development that I will describe in the following paragraphs.
With Fellowship One's currently funcationality, a "group" is formed by an individual's assignment to an Activity. What we are building is a fundamental shift in philosophy and funcationality. An individual is a in a group because you say they are that group, nothing more. As it exists today, the assignment is the link between an activity and the individual, thereby forming a group of people. What we are doing in the new is not requiring there to be an activity for the group to exist. So, a group can be "Prayer Warriors"___ individuals at your church that don't necessarily meet at a specific date or time (activity) but are nevertheless a Group.
In future development, we will allow you build "Events". Events can stand on their own, or be married to Groups. This will eventually take the place of the existing activity structure.
Understanding what Groups a person belong to will be an integral aspect of Fellowship One.
Scott