If the heater doesn't fire at all, and the fans don't run, the thermostat and filter should first be checked and repaired if necessary. It's also worth checking connected thermometers; improper readings can cause the system to not start.
Thermostats typically have a "fan" setting that bypasses other components of the system. If the fan doesn't run in this mode, the problem may lie with the electrical components of one of more of the fans. Replacing a fuse can sometimes fix the problem, as can resetting a circuit breaker. Wires degrade over time, so homeowners can check for signs of wire damage. More extensive work typically requires expert care, but homeowners with some electrical experience can check the wiring to the control board, blower motor and other easy-to-access parts.
If the fans run, but the system doesn't produce heat, the electrical components should first be inspected to look for obvious signs of problems. If none are apparent, it's worth taking a look at the heating element, which uses electrical resistance to generate heat. If the system is on, and the heating element produces a small amount of heat, it might need to be replaced. Removing the unit might also show signs of damage as broken wiring can render the entire unit unusable.