Air handlers and furnaces aren't typically used side by side in North America. If you have a furnace, you probably won’t need to think about an air handler.
Air handlers are frequently paired with heat pumps and manage airflow across your residence. Some systems also provide backup heating and cooling pieces to support the heat pump.
A furnace runs in a different manner by sending warm air into your vents, which send it around your house. Since furnaces depend on combustion chambers to create heated air, they don't have some of the parts you'll spot in a regular air handler.