A zero cross relay will turn on when the AC supply approaches 0V. The zero cross can vary in the actual voltage it operates at. it can be between 0V and ±20V depending on restrictions in the circuit configuration.
The reason for this is to prevent noise being generated in the supply voltage. If a relay turns on in the middle of the AC wave it will cause the voltage to drop creating noise in the voltage supply line because of the very sudden large current change.
It also helps to prevent large inrush currents which can be damaging to certain devices like resistance heating elements, ballasts, lamps, and any device with low impedance at startup.
Here is a quick drawing of the zero cross relay operation.