How can I control a 120v, 1amp fan with a Nest thermostat?
You need a 24 VAC transformer to power the thermostat, and a 24VAC (coil) 120VAC (or more) (contact) 1amp (or more) (contact) relay.

The 24VAC supply goes to the thermostat and one side of the relay coil. The other side of the relay coil goes to the switched terminal, and the (24V supply) neutral goes to the thermostat. 

The hot supply of the 120VAC goes to one side of the relay contact, the fan goes to the other, the (120V supply) neutral and ground go to the fan.

When the thermostat switches, the relay coil energizes and your 120V load turns on. 







simulate this circuit &ndash; Schematic created using CircuitLab