DIAGNOSIS The control board is getting power (hence the light), but it's not sending the signal to start the cooling cycle. This often happens if the unit was just plugged in and is in a 'start-up delay' mode, or if the internal thermostat probe is faulty.
FIX Unplug the cooler from the wall. Wait 30 minutes. This resets the control board completely. Plug it back into a dedicated outlet (not a power strip) and set the temperature to the coldest setting (usually 34°F). Listen closely near the back bottom—you should hear a faint click and a hum within 5 minutes. If you do, it's working.
IF THAT DIDN'T WORK The thermal fuse on the compressor might have blown, which is a safety device. This requires checking continuity with a multimeter. If you're not comfortable opening the back panel, this is a service call.
HEADS UP These coolers are sensitive to being laid on their side. If you just transported it, let it stand upright for at least as long as it was on its side before plugging it in, or you can oil-starve and lock the compressor.