There are two primary types of ASHPs:
Ductless air-source heat pumps are exactly as they sound: heat pumps that don't require that you have ductwork in your home. Each ductless system includes one outdoor unit connected to one (single-zone) or more (multi-zone) indoor wall, floor or ceiling air distribution units. Ductless ASHPs are often referred to as ductless mini-splits (if one indoor unit is connected to each outdoor unit), or ductless multi-splits (if two or more indoor units are connected to each outdoor unit).
Ductless air-source heat pumps can be installed as a primary source of heating and cooling or installed to heat and cool specific rooms. This could include, for example, installing ductless units in the most frequently used rooms like family rooms or master bedrooms to displace heating or cooling from your existing system, or placing ductless units in rooms or new additions that never seem to be warm or cool enough.
These systems can be used for heating, cooling, dehumidification or as a fan. Because each indoor unit can be controlled individually, you can reduce your energy use even more by lowering the temperature in rooms that are not being used.
Ducted (also known as central or unitary) air-source heat pumps have an outdoor unit that is connected via refrigerant lines to an indoor air handler that uses a building's ductwork to distribute heating or air conditioning throughout the home. Note that not all ductwork is sized adequately for heat pumps, especially in older homes. Heat pump installers can tell you if your ductwork is sized adequately and what modifications may be necessary.
Regardless of whether a system is ductless or ducted, all ASHPs will have an outdoor unit (pictured below), which will be mounted on a ground platform or on the side of your home.
Above: A ductless ASHP outdoor unit; Below: A ducted ASHP outdoor unit.
If you’re installing a ducted ASHP, the outdoor unit will be connected to a central air handler similar to one used by a furnace or central AC system. If you’re installing a ductless ASHP, the outdoor unit will typically be connected to one or more wall-mounted units (pictured below).
For homeowners that might not have suitable wall space or don’t like the aesthetic look of the wall-mounted indoor unit, floor-mounted (below) and ceiling-recessed units are also available, though these units cost more to install. 
Photo courtesy of E. Armstrong