What is the purpose of the heat exchanger on my home's A/C unit?
i think what you mean is the air exchanger,if you have one of the new ac units,they come with a air exchanger to bring in fresh outside air.
If you really mean heat exchanger it is part of the furnace, it has the burnt gasses on the inside and the air the heats the house goes around the outside, it keeps the carbon monoxide from getting in the house, if it is broken it is life or death, GET IT FIXED OR REPLACED.
If it is an outside unit ... the heat exchanger removes heat from the compressor coils. This works just like a refrigerator. As the compressor works to lower the temperature of the refrigeration liquid, it must remove the heat. The unit (this is how ours works) has a fan that sits in the middle of a housing ... the housing is surrouded by metal tubing. The fan helps to blow air across these tubes to cool them. One of the reasons it's important to keep these units clean (leaves, yard debris, etc), is so the heat is removed efficiently. This allows the unit to operate efficiently and ultimately saves wear on the compressor and energy.