George:
You are correct, the nature of an API is to act as a tool to build applications on. By itself without anything actively consuming it, it will lay dormant. Code must be written. The only thing you can use from the REST API without code is the built-in documentation.
With that, you can have batch processes or jobs that can consume the API without any user intervention - but in that case the batch application is acting as the user.
A PULL API (such as Data Exchange or the REST API) is like a gun - no bullet comes out (data) unless the trigger is pulled (a request).
It sounds like what you're referring to above is more like a PUSH API; much like how the enterprise services for iPhone work. Where email, contacts, etc... can be pushed to the phone from the server.
So, while some Google services have PUSH capabilities (just mentioning that because it seems that it came up in your conversations) you would still need a catcher’s mitt (a receiving application) to put the data into Fellowship One.
Addressing your need for a layman's description to pass on:
How: Check out the videos the developers created based on the apps that were demoed at DC09. They addressed real ministry needs - I think the Facebook video probably is the best illustration if you we're talking to youth pastors, iServe does a good job for Volunteer leaders, and the Expression Engine video is solid when you're addressing Web developers and even high levelers that are looking for a quick win on their website.
Who: For the REST API, we are hoping everyone, but knowing that you need a static perspective - I would say Churches, 3rd parties, and Fellowship One. Still not clear? I mean that if anyone wants to share their application with other Fellowship One customers, all they have to do is "share it", and if anyone wants to use shared applications all they have to do is opt-in.
The API will allow a developer to build / extend features of your web presence and backend processes and allow your church to use features built by others from the community on the REST API.