There are several different types of relays in your Heating And Cooling
System.  Those that can be replaced as a single component and not
soldered onto a control board usually cost between $16 - $35 and can be
replaced in 20 minutes or less.  Total Job:  $48 - $85  
Phone: (757) 506-6488

They do this by controlling the voltage to the component through a
set of contacts.

The thermostat will send voltage through the relays coil and it will
make or break the set of contacts sending, or stopping, current flow.

applied it takes a minute or two for the fan or component to come on
and likewise, when the voltage stops there is delay in shutting down
the component.

All of these relays can be replaced as a single component and there
are just a couple of screws to take out and a few electrical
connections to make.

Sometimes the electrical connectors have to be replaced but this
should be at a minimal cost