Zero Fuel Weight is the weight of the aircraft with everything loaded for that flight, minus useable fuel (Takeoff Weight minus weight of useable fuel).
The Zero Fuel Weight in your weight and balance calculations are just a reference mark. It is not used for much if anything in most flight applications since you should never get to within 30 minutes of your empty status. This reference is useful in calculating other reference points. One of which is your maneuvering speed (Va). If you know your Va at your takeoff weighting your Va at your Zero, it gives you an envelope to remain within when it is necessary to slow down to Va. Slowing down to your Landing Weight Va speed during turbulence is your best option.
Your Center of Gravity calculation is just as important. Keeping Takeoff, Landing, and Zero Fuel CoGs in the envelope ensures best performance for things such as spin recovery and stability during all phases of flight.