A Heater Core (also called a Heater Matrix) is the unit within any vehicle equipped with heating controls. The basic concept of this unit (and the reason why you don't get hot air immediately when your car starts on a chilly day) is a miniature radiator that runs a mixture of antifreeze (called coolant)through its fins.

This coolant is fed off of the main cooling system via individual valves that prevent a blockage from within this tiny radiator from affecting the rest of the cooling.
When you switch your Heater on, and the engine is warm, a blower motor pulls air from outside or from inside the car, forces it through this radiator, which heats up the air, and out through the vents. Switches like defrost and vent and floor vent control small doors near this core, which direct the air flow to different aspects of vents in the dash.

This heater core is actually located in the dash, right next ot the firewall on the passenger side (or in the engine compartment, or behind the center of the dash). This device is one of my major pet peeves. When they leak (and they do leak) it can be as bad as a huge dark stain in your carpet from leaked fluid, you'll definitely smell it, steam will most likely come from your vents (even if you don't use the heater) and it should start itching or burning your face with enough time.

Breathing in this stuff is not fun. Worse yet, if the thing happens to bust, it is one of the hardest items to change on a sports car. It was my first time changing a heater core on this 91 Camaro Z28 Convertible; I spent two days, was unable to completely disassemble the dash, and finally with a little luck, got it out, only to have more problems putting the new one back in.

On an additional note; Being an additional radiator, if your vehicle is overheating for any reason, these cores will help cool the vehicle off. Just roll down your windows, turn on the heater full blast, and cruise along. It may not be a significant help, but I've used this technique before, and it keeps you off the side of the road.

The core itself ran $16. A shop charges normally between $175 and $250 for a job like this. Toys under the dash are not fun.