A heat pump water heater — often sold as a hybrid — has a small refrigeration circuit on top of a conventional tank. It moves heat out of the surrounding air into the water rather than generating it with an element.

The efficiency
Electric resistance water heaters have a UEF around 0.90 to 0.95.
Heat pump water heaters have a UEF around 3.0 to 4.0, meaning three to four times the hot water per unit of electricity.
That is not an incremental improvement. For a household on electric water heating it is the largest single running-cost reduction available in the house.
The conditions
Air volume. The unit needs a supply of air to extract heat from, and manufacturers specify a minimum space — commonly on the order of 700 to 1,000 cubic feet — or ducting to and from a larger space.
A heater in a small closet with the door closed will run out of heat to extract and fall back on its resistance elements.
It cools and dehumidifies the space. In a basement in summer that is a benefit. In a conditioned utility room in winter it is a load on the heating system, and the heat it "saves" partly comes back out of the furnace.
It makes noise, comparable to a refrigerator or a small air conditioner. Beside a bedroom that matters.
It produces condensate, which needs a drain or a small pump.
It recovers more slowly than a resistance heater in heat pump mode.
The hybrid modes
Most units offer several operating modes:
Heat pump only — most efficient, slowest recovery.
Hybrid or auto — heat pump normally, with the elements engaging under heavy demand. The usual setting.
Electric only — resistance, for a period of unusually high use.
Vacation — a low setpoint while away.
A household that finds itself running in electric mode most of the time has either an undersized tank or an inadequate space, and it is getting none of the benefit.
Where they fit well
An unconditioned or semi-conditioned basement, which is the ideal location: plenty of air, and the cooling and dehumidification are welcome in summer.
A garage in a mild climate.
Any household replacing electric resistance water heating, where the running cost difference is largest.
Where they fit badly
A small closet without ducting.
An unheated garage in a cold climate, where there is little heat to extract and the unit spends the winter on its elements.
Beside a bedroom, for noise.
Where recovery speed is critical and the tank cannot be sized up to compensate.
Sizing
Because heat pump recovery is slower, the usual approach is to go up a tank size relative to a resistance heater — the stored volume compensates for the slower reheat.
Compare first hour ratings rather than gallons, as with any storage heater, and note that the published FHR may assume hybrid operation with the elements available.
Cost and incentives
The equipment costs substantially more than a resistance heater, and it is one of the most heavily incentivised appliances there is — federal credits, state programmes and utility rebates, with income-qualified programmes covering most of the cost in several states.
With incentives applied the payback against resistance heating is frequently short, and the requirements are worth reading before choosing a model, since programmes list qualifying units by UEF and by model number.
Maintenance
The tank side is conventional: an anode rod to check every few years, and a periodic flush for sediment.
The heat pump side adds an air filter on the intake, which needs cleaning every few months. A blocked filter is the most common reason a unit falls back on its elements, and the owner is usually unaware there is a filter at all.
The condensate drain needs checking on the same schedule, since a blocked one will shut the unit down or, worse, overflow.
Ducting
Where the space is too small, most manufacturers permit ducting the intake, the exhaust or both to an adjacent larger space or to outdoors.
Ducting the exhaust outdoors in summer removes the unwanted cooling; ducting the intake from outdoors in a cold climate is generally counterproductive, since the unit has less heat to work with.
The duct lengths and diameters are specified, and exceeding them reduces performance the same way it does on any air-moving equipment.