A, B, and C are the names used by Airbus. As a short summary:
If you want text communication with ATC (CPDLC), you need FANS A in USA/oceanic, and FANS B or C in Europe.
If you want position/trajectory reporting at set intervals (ADS-C) in continental Europe, then you need FANS C (still being tested).
See the Airbus slide below for more: