The coil of the relay will have (probably) 24vac on it if it should be energized. ALso, the input to the relay should have line voltage on it (it could be either 120vac or 240vac, depending on the type of motor). On many furnaces, there's an internal thermostat that looks at the heat exchanger temp, and turns the fan on when that gets hot. So, it could be an internal fan control thermostat. Does the burner come on, then go off? If it comes on, how long does it run before it shuts off?
There are so many things that could be preventing it from running that without more info, it's impossible to say. If you have the manual(s) that came with the thing, read the theory of operation, then follow the sequence of operations...see where it stops. Then, follow the control voltage. When the thermostat is turned on and off, do you hear a relay in the furnace energizing (it might not have a mechanical one, it could be electronic)? the thermostat on your wall may no longer be working, and it may not be sending the signal to the furnace. That's fairly easy to check IF you know what you're doing. All the thermostat is is a switch. WHen it calls for heat, it closes an internal switch that applies 24vac to the furnace to energize it into heating operation.
A HVAC supply store would likely have a compatible relay and they usually aren't all that hard to replace, but it's a waste of time and money until you verify it's actually being told to energize - the relay could easily be fine, and there's some other system fault. Modern furnaces have lots of safety interlocks, and it could just as easily be a safety switch that is no longer working, or it has actually detected a fault, and is preventing the furnace from running.
So, more info is needed.