[1] This is in relation to solid-state relays, such as TRIACs and silicon controlled rectifiers.
[1] The purpose of the circuit is to start the TRIAC conducting very near the time point when the load voltage is crossing zero (at the beginning or the middle of each AC cycle represented by a sine wave), so that the output voltage begins as a complete sine-wave half-cycle.
In other words, if the controlling input signal is applied at any point during the AC output wave other than very close to the zero-voltage point of that wave, the output of the switching device will "wait" to switch on until the output AC wave reaches its next zero point.
This is useful when sudden turn-on in the middle of a sine-wave half-cycle could cause undesirable effects like high-frequency spikes, for which the circuit or the environment is not expected to handle gracefully.
A zero-cross circuit works to correct this problem, so that the TRIAC functions as well as possible.