Mini Whirlpool Refrigerator
Refrigerators turn on / off based on the internal temperature, check your thermostat dial and make sure it is set properly - perhaps initially to coldest. If it has a power switch as you said - model number not listed, cycle the power switch.

Some of these units have a trip on the back side of the unit to protect the compressor - check that it is not tripped (might be a red or black push pin style), light will work but cooling will not, it protects the compressor it is an overload, unplugging and plugging while running may have caused this to trip.