What are the differences between FANS A, B, and C? Is FANS C commissioned?
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).



  
  (Own work)


See the Airbus slide below for more:


  
  Source: Airbus via icao.int (PDF)


Operational demonstration of FANS C started a year ago in Europe.

Related:


How does ADS-C and FMS relate to FANS?
What&#39;s the relationship between FANS and CPDLC?




Acronyms list:


  ► A623: ARINC protocol defining a set of data
  link applications (DCL, OCL, D-ATIS)
  ► ACARS: Aircraft Communication and
  Addressing Reporting System
  ► ACR: Avionics Communication Router
  ► ADS-C: Automatic Dependent Surveillance -
  Contract
  ► ADS-B: Automatic Dependent Surveillance -
  Broadcast
  ► AFN: ATS Facility Notification
  ► AOC: Airline Operations Control
  ► ATN: Aeronautical Telecommunication
  Network
  ► ATSU: Air Traffic Service Unit
  ► CM: Context Management
  ► CPDLC: Controller Pilot Data Link
  Communication
  ► CVR: Cockpit Voice Recorder
  ► D-ATIS: Digital ATIS
  ► DCL: Departure Clearance
  ► DR: Data link Recording
  ► FANS: Future Air Navigation System
  ► FMS: Flight Management System
  ► HFDL: HF Data Link
  ► HFDR: HF Data Radio
  ► IMA: Integrated Modular Avionics
  ► LRU: Line Replaceable Unit
  ► OCL: Oceanic Clearance
  ► RMP: Radio Management Panel
  ► SDU: Satellite Data Unit
  ► VDL: VHF Data Link
  ► VDR: VHF Data Radio