DIAGNOSIS That's the internal relay clicking on and off. It's normal when the thermostat is calling for heat or cool and your HVAC system isn't responding.
FIX The Nest is sending the signal, but the system isn't turning on. Check your HVAC system's power. Ensure the furnace/air handler breaker is on and the service switch is flipped up. Also check the condensate overflow switch if you have an AC—if it's full, it will cut power.
IF THAT DIDN'T WORK There may be a wiring issue. The Nest's relay clicks because it's trying to complete a circuit that's broken. Pull the Nest off the base and inspect the wires for the 'Y1' (cooling) or 'W1' (heating) terminals. A loose wire here causes this.
HEADS UP Continuous rapid clicking can wear the relay. This is a sign your HVAC equipment isn't receiving the signal. The problem is almost always at the furnace/air handler, not the Nest.