Let's say I have equipment of resistance $R$ (maybe a bulb or heater). If I apply a voltage $V$ to it, the power used by the equipment will be $V^2/R$ assuming Ohm's law holds for the equipment.
If I change the voltage, the power used by the equipment changes.
But now, I have equipment with a power rating that says $P$. Now; if I apply voltage $V$ to the equipment, I get a current $I$ given by $I = P/V$. But now, if I change the voltage, the power won't change; instead, the current will change.
Why is this so? So how is the power across the equipment constant in the second case? Is there something else that I am missing here?