Seems like they have heatpump/boiler systems. So, the heat pump provides heating and ac and the boiler provides supplementary heat, instead of the the electric strips a heat pump system usually uses. (Heat pumps have trouble supplying heat below 40 degrees F.) Presubably there’s a coil inside the heat pump that tranfers heat from the hot water to then air.
An air handler is a cabinet with a fan in it. It will also have coils from the boiler and from the cooling system. The air handlers blow air past the coils; this heats or cools the air, which then travels through the ducts to the room being heated. The cooling could either be from a heat pump or an air conditioner.