I am not very knowledgeable about ADS-C (I'm more of an ADS-B wizard), however, my understanding is that both ADS-A and ADS-C communications are carried out via the ACARS system, whereas ADS-B communications are emitted from the aircraft via either the transponder (1090 MHz method) or, in very small aircraft in the United States, via a UAT (Universal Access Transmitter).
There is a fairly comprehensive article on Wikipedia that explains ACARS, at this link:
Aircraft Communications Addressing and Reporting System. There is also a fairly comprehensive article on Wikipedia about ADS-B at this link:
Automatic Dependent Surveillance-Broadcast.
I think if you read the ACARS article first, then the ADS-B article second, you will be able to get a pretty good understanding of the differences between the ADS-A and ADS-C systems and the ADS-B system.
The three ADS systems are all mutually independent, you don't need one in order to use the other.
Michael