Depending on what power you need for the fan to work , connect the fan black ground to negative / ground -ve of the power supply
And connect the fan positive to NO or NC depending on which way you want it to work,
And then common to the power supply positive +ve
So that would.operwte the fan when the relay is triggered, if it's. Magnetic gate then that power arrangement would probably be setup via the NC route , or you won't use any power if there is an existing push button which you can wire across it without any power arrangements
Or you will be probably blow it all up if your not understanding it correctly ?