AUX HEAT means auxiliary heat and EM HEAT is emergency heat, sometimes also abbreviated as EMERG HEAT. Both settings turn on an electric resistance heat strip in a heat pump air handler, which is similar to the heating element in a toaster oven, but they activate it in two different ways.
To understand why an electric resistance heat strip is needed, it’s necessary to review how a heat pump works. It does not actually create heat but, instead, absorbs heat from one location and moves it to another. During the summer it absorbs heat inside your home and moves it outside, with a reverse in the direction of heat transfer happening in the winter and heat moving from the outside to the inside. When the outdoor temperature gets close to freezing, it becomes difficult for a heat pump to efficiently absorb heat outside to transfer into the home. If the heat pump needs assistance to maintain the indoor temperature—either because the thermostat has been been set more than a few degrees above the current room temperature, or because it is unable to maintain the thermostat setting with the heat pump alone and is falling behind—it automatically switches on the electric resistance heat strip to produce additional heat. Typically, a small light turns on next to the words AUX HEAT on an older thermostat like the one above, or AUX HEAT appears on the screen of a new digital thermostat.