SEER stands for Seasonal Energy Efficiency Ratio. It is a measure defined by the Air Conditioning, Heating, and Refrigeration Institute to rate the cooling efficiency of air conditioner or heat pump. Let me emphasis that again, the SEER rating only reflects the cooling efficiency of a heat pump, not its heating efficiency. Heating efficiency is rated using a different metric called HSPF. See this post for more information about HSPF.
The efficiency of a heat pump is highest when the difference between the indoor and outdoor temperature is the least. For this reason, air-source heat pumps are much more efficient in the spring and fall then in the winter and summer. The SEER rating attempts to take seasonal variations in heat pump efficiency into account. For this reason, it can be thought of as an average efficiency over the course of a year.
SEER is defined as the ratio of the thermal energy transfered out of your home per cooling season in British thermal units (BTU) divided by the electrical energy consumed by the heat pump in watt-hours (Wh) per cooling season. More compactly,
SEER = (thermal energy transfered, BTU)/(electrical energy used, Wh)
As a physicist, it doesn’t make much sense to mix two units of energy (BTU and Wh) in the same formula. It makes understanding the formula more complicated. If one notes that
1 Wh = 3.412 BTU, then
SEER = 3.412 (thermal energy transfered)/(electrical energy used)
where the energies can be in any units so long as they are the same units. In this form, you can see that a 100% efficient air conditioner has a SEER rating of 3.412. Thus a heat pump with a SEER rating of 6.8 would remove about twice as much thermal energy from your home as it consumes in electricity.
Although the SEER rating you will actual achieve depends on your climate (hotter climates will achieve lower real-life SEER ratings than cooler climates) replacing a 10 SEER heat pump with a 16 SEER heat pump will decrease the amount of electricity needed to cool your home by about 37.5%. Replacing a 10 SEER heat pump with a 22 SEER model will reduce the electricity used by your heat pump by about 54.5%. So there is diminishing returns to energy savings with increasing SEER rating. In general, you can calculate the percent reduction in electricity used by a new heat pump to cool your home using the following formula.
Percent Energy Savings = 1 - (Old System SEER)/(New System SEER)