"Air handler" is a generic term for the unit that contains your indoor blower and has direct contact with your indoor air. With commercial units, the air handler may or may not do anything except direct air between sources and zones calling for heating or cooling. With residential systems, the manufacturers tend to pack whatever they need or want to into the air handler unit.
In a heating-only system, the unit is usually called a "forced air furnace" if it runs off electric resistance elements or a gas burner. If it has a water coil and runs off a separate boiler, they call it the air handler. If it is a heat pump, they call it the air handler.
In home systems that cool, most a.h. units are classified as split or packaged. Packaged systems contain the whole system (except maybe the thermostat), so the only thing that connects indoors is ductwork for supply and return air. Split systems have a separate outdoor unit which contains the refrigerant compressor, outdoor heat exchanger and blower, as well as some control circuitry. With split systems, there are usually no ducts leading to the outdoor unit. The indoor unit may have a stack vent built into it, which indicates it has a gas furnace built into it.
In your situation, a retail product that does what you want to will be called a "chilled water air handler". It will consist of a blower, a radiator of some sort, and a drain pan. They make a jillion bagillion different designs, which may (or may not) contain a pretty cover or box enclosure, ductwork, thermostat controls, pumps or valves, etc.
Basic uncased unit:
Typical closet unit:
Ceiling cassette:
