How does the heater core work?
What are the inputs to the system?
What are the outputs of the system?
What are some common ways that it can fail?
The heater core is part of continuous loop of engine coolant propelled via the water pump, flowing through caverns in the engine, through pipes that run through the firewall into the heater core, back out the firewall, and finally through the radiator. If the thermostat dictates it (due to a temperature limit being reached), the fan blows air over the radiator to cool it.
Since the liquid flowing in and out of the heater core is constantly heated by the engine, the only other component necessary to produce heat is a fan blowing air through the heater core (much like the radiator, except the air enters the cabin.)
So, what can go wrong:
The heater core works through heat transference. Hot radiator fluid goes into the core, the heater fan blows through it, heat is transferred to the cooler air, and heat is transferred into the cabin. The input is radiator fluid directly from the engine and output from the core goes directly back into the engine. It is moved through via the water pump on the engine. Common failures are through being plugged by debris in old radiator fluid, by corrosion, and to a lesser degree, through the heating and cooling cycles which can cause failure (through loosening joints and such).