Boilers for comfort heating can provide either hot water or steam through piping. Chillers for comfort cooling can provide chilled water (technically a water and glycol mixture to prevent freezing) through piping.
Using the steam/hot water/chilled water for space heating and cooling can be done in a few ways. This will be a very simplified.
Heat
Direct radiation/convection through the use of terminal units. Think of this as how a typical home would heat if a boiler is used. Radiators out in the space, or baseboard style fin and tube. There can also be buried piping for in-floor radiant heating as well (hot water only)
Heat and cool
Through the use of water to air heat exchangers (coils) you pass air over heated/chilled water coils to pick up/dump heat into/from a space. Typically used to temper large areas. More often than not a mixture of return (space) and outdoor air is used. These units are called Air Handlers.
If outdoor air is within a specific range during the summer/cooling, you can also “economize”. This is using outdoor air to cool/dehumidify a space. This is dependant upon the difference in enthalpy between indoor and outdoor air. Typically the.morning or at night is the only time a unit will economize due to lower humidity and temperature levels.
All furnaces are air handlers, but not all air handlers are furnaces.
Furnaces combust gas (natural gas, LP, oil) and pass air over a heat exchangers to deliver heated air. Not all air handlers have a section for combustion, but all furnaces do.
A normal split cooling system has two coils, an evaporator and a condenser. The evaporator is the coil in the indoor air flow and the condenser is placed outside. Here is a typical refrigeration circuit

There are also heat pumps and VRF which is likely what you will be running into more and more often these days. Internals on a heat pump look like this

A split system is what I just described- an indoor air handler with an evaporator, line set (refrigeration circuit) piping in between, and a condenser outside. For heating, the air handler could be a furnace, or it could use a hot water coil, steam coil, heat pump, electric resistance heaters etc etc.
I will list a few different ones so you can see the scope of how different things can be…
1). 9 story apt bldg downtown. Geothermal central heat pump loop water pump plant in basement (Variable frequency drive- basically modulating speed pumps based on differential pressure between supply and return). Pump loop is fed up through building to each apt with zone valves (which requires vfd driven pumps. As zone valves close, pressure differential increases- pumps slow down) tied into a heat pump. Depending on mode (heat/cool) heat pumps either pick up or reject heat to heat pump water loop (more.often than not, heat pumps are in the same mode depending on season).
As a “for instance”, let’s say it’s august and all heat pumps are in cooling. The water loop is running at 60 degrees without load. As heat pumps reject heat into the loop it warms up, and our design temp is 66 degrees. A mixing valve begins to open as temps reach 66 degrees in the loop and we start feeding water down into the geothermal loop where heat will be rejected. A geothermal loop is a system that sends piping down into groundwater for heat rejection/absorption. Groundwater will stay at the same temperature year-round. In summer, the heat pump water loop is rejecting heat from the space(s) into the ground. In winter, heat is absorbed into the water loop to be used by the heat pumps in the space.
2). School with existing steam system and vacuum return. Central boiler system piped and zoned to each individual classroom with pneumatic thermostats and pneumatic valves. As a room needs heat, it takes from the pressurized steam main through a valve allowing steam into a unit ventilator. As the room meets temp, the valve shuts. Condensate return is tied into a vacuum pump allowing for faster condensate return and potentially lower steam temperatures depending on load.
Cooling is done by individual cassettes built into unit ventilators.
Gym heat/cool load is taken care of by a roof top unit. This unit is ducted in through the roof and has a gas fired heating section and cooling section.
Auditorium is conditioned by a split system, heating taken care of by a steam coil.
STEM wing is individually zoned with VRF system which takes care of the cooling load and majority of the heating. Bathrooms, break rooms and high demand areas supplemented with hot water from heat exchanger piped into steam system. Make up air unit takes care of 50 foot ceilings in main foyer and replaces air that is exhausted in chemistry rooms.
3). Multipurpose bldg (retail). Each space is controlled individually by gas fired furnaces and split systems or roof top units. Central high efficiency hot water boiler for common areas and public restrooms fed into hot water coils/air handlers.
Sorry this is such a mishmash of info dude. I tried to be descriptive but brief. Please let me know if there is anything else I can do to try to explain some of this. If you’d like I can recommend a few books to get a bit more info on this stuff, steam and hot water specifically.