The air conditioner in your home, should you be so fortunate as to have one, is a heat pump that only goes one direction. It takes heat from one place and puts it in another, from inside to outside. If you're one of those people with a heat pump instead of a regular furnace, then your machinery can change directions and move heat from outside to inside. It doesn't have to reverse the direction of the compressor to make this happen, it's done with a valve the changes how the refrigerant flows from the compressor to the indoor and outdoor coils. Whichever coil comes first after the compressor is being heated; outdoor first is moving heat from inside to out: air conditioning. Indoor coil first means heat is being moved from outside to in: heating.
Of course when it's cold out you want heat to be moved inside, but you should be so lucky. It's cold outside, right? There isn't that much heat to move.
That isn't entirely accurate. On the Kelvin scale there's lots of heat to move, but using the refrigerants - the fluids that carry the heat from here to there that are readily available, the lower end of the Kelvin scale is essentially off-limits. It takes more energy, higher pressures, etc.
Why pressures? You might not know exactly how the heat pump works, but remember when you were a kid, pumping busily on the bicycle pump to get some air into your flat tire? Yeah - and did you ever feel the base of the pump? It gets warm. Pump up something bigger and it can get downright hot. You take a compressible fluid - like air - and squash it down into a smaller volume and the heat energy in its original volume now has to manifest in a smaller volume. It does that by feeling hotter. The amount of energy represented is the same. Imagine: LEGO blocks on the floor represents the ambient temperature. Make them take up less space by stacking them: now they're taller and whatever's above that ambient level, you can peel off and use elsewhere - very simplified, that's what the heat pump does. The heat pump squashes the gas into a smaller space, and stacks the heat higher. There's still the same number of blocks, but you can peel the higher ones off and send them elsewhere, by blowing the heat off with a fan. Allowing it to expand in the evaporator coil lets the fans blow the heat off the refrigerant, either into your house or into the great outdoors, depending on what you're doing. The hot side is compressed so it's warmer than the air around it, and extra heat can be blown off. The cool side is cooled below the air around it, so it can absorb heat from the air.
I'd send you to an animation explaining heat pumps, but if you're not interested in the subject, they're really not helpful.
There's another limiting factor, too - water, or at least the presence of it and its pesky freezing point. There's a lot of water in the air, especially in the humid climes of East Tennessee, and running the heat pump to pull heat from outdoors to indoors means cooling those outdoor coils well below the freezing point of water. That means water vapor in the air freezes directly on the coils as frost. Next thing you know, there's two inches of frost on your coils, no air is going through them at all, and the silly thing just keeps running because the idiot thermostat continues to call for more heat. There's no more heat. No air going through the coils to give up the heat to the house.
If it's such a pain to have a heat pump, why have one? Well, if you're not on a gas line - and I'm not - then it's the next cheapest way to heat your home. Running resistance heat, sometimes called strip heat, gives approximately a 1:1 ratio, one kilowatt-hour of electricity equals one kilowatt-hour of heat. It's not perfect, there's losses in any system. But fire up a heat pump and you're not trying to make heat, you're just moving it around. That takes more complex hardware, but it's more efficient. Even my crusty old heat pump is probably performing at close to a 1:2 ratio, one kilowatt-hour of electricity is good for two kilowatt-hours worth of heat.
That's lousy performance, by the way. A coefficient of performance (COP) of only 2.0 is just terrible. If it were a new unit doing as poorly as that, I'd tell the installer to come and take it back. New units can do much better, quieter, etc. If you have a ground source heat pump, which dumps waste heat into the ground, and picks up conditioning heat from the ground, you can blow past a COP of 4.0 with little difficulty; COPs in the 5.x aren't unheard-of. A bonus there is that the outdoor coil, buried as it is, doesn't frost over. Ground source units keep working as long as the temperature of their ground coils stays relatively warm. Around here, a coil 300 feet long and buried about four feet deep would be sufficient for a house my size, with a nice margin for extra-cold days.
So how do new heat pumps keep working when the weather chills? Well, there's multiple stages of performance. Full speed on the compressor and fan for mild weather. Low speed on the compressor and full speed on the fan when it gets cold - more air mass moving through the coils while the compressor pulls less heat from the air, so it doesn't cool the air too much and frost the coils. Defrost cycles that briefly warm the coils up, drive the moisture off them so they can keep working. And of course there's "Emergency Heat," the resistance heating elements. Think of a really big space heater, plugged in and glowing: that's Emergency Heat. Sometimes you'll see a heat pump paired with a gas or oil furnace, too. When the snow flies and mercury drops, the flame fires up and you're still toasty.
Man, don't I wish. When my place gets cold, the wood stove comes into play. It gets the job done, too, but its off at one end of the house, so we tend to hang out in that one room.
Fantasy Heat Pump
I like to imagine an installation of solar panels on my roof that's heating a large holding tank of water below my house. It raises the temperature of the water to something comfortably above ambient, maybe as warm as 70-80 degrees in the middle of winter. This would be a large tank, over a thousand pounds of thermal mass. And instead of the heat pump using an air coil, it would have a water coil dipped into the tank. It would pull stored heat from the tank, squash the refrigerant and blow the heat off, into the house. In the summertime, disconnect the solar feed so it isn't heating the water, and the air conditioner cycle dumps heat into the cool water. If nothing else, this eliminates the need for the noisy outdoor fan and coils, buzzing and humming outside my bedroom window all night.
Fantasy Air Conditioning
I've also wondered whether I really need the very cold air the air conditioning cycle generates. I don't cool my house very much; were I to have a ground source coil in the ground, couldn't I just run that coolant directly through a heat exchanger? The temperature would be about 58 degrees, which certainly feels cool enough in our hot summers. I don't cool my house much below about 78 degrees anyway, and all this would need would be a small circulating pump and a fan. That's way less power draw than the big AC compressor.
You won't see that many people admitting that they fantasize about heating and cooling schemes. I'm man enough to step up and claim it.