The power rating of an appliance tells you how much energy it transfers each second when it is working normally. Power is measured in watts (W). One watt is one joule per second.
A kettle rated at 2000 W transfers 2000 J every second from the mains supply to the thermal store of the water. A hairdryer rated at 1500 W transfers 1500 J each second to the thermal store of the air and to the kinetic store of the fan and the air. A 60 W lamp transfers only 60 J each second, mainly to the thermal store of its surroundings and as light.
A higher power rating means more energy is transferred each second, so the stores change faster. The total energy transferred is found using energy = power × time. A 60 W lamp left on for 10 s transfers 600 J.