In the first place, forget frequency, and understand the relationship between voltage V in volts, current I in amperes, and power in watts.
Power in watts at any instant in time is voltage in volts times current in amperes, or $I(t)V(t)$.
If you have an alternating current power source, the voltage and current can change from one instant to the next.
Added: OK, let me try to say a little more about it, in informal terms.
On the one hand, suppose you have a big powerful battery, like a 12-volt car battery.
It wants to supply direct current (DC) - its frequency is zero (0).
Now you connect an incandescent automobile light bulb to it of, say, 120 watts (a really bright bulb).
For that light bulb to dissipate 120 watts of power, at 12 volts, it has to draw 10 amps of current.
(By the way, to do that, it has to have resistance of 12v/10a = 1.2 ohms, but that wasn't your question.)
Now, reverse the battery connections to the bulb.
It still is as bright as before, still dissipates 120w.
The difference is, the voltage across the bulb is the opposite of what it was before, and the current is the opposite also, but when they are multiplied, the power is still positive.
Suppose you reverse the connections 60 times per second, or 120, 400, or a million.
How much power does the light bulb dissipate?
The same, because there is no time when V times I is not equal to 120w.
Typical wall power is not like that, of course.
Rather than instantaneously switching between positive and negative voltage, it swings smoothly with a sinusoidal curve.
So the light bulb dissipates maximum power when the voltage is maximum in the positive direction, zero when the voltage is zero, and maximum again when the voltage is maximum in the negative direction.
So if the wall frequency is 60hz or 50hz (or 400hz in some airplanes), the power dissipation actually varies between maximum and zero at a rate of twice that.
But since the filament takes a longer time to get hot and cool down, it seems to be on steadily.
The running average power is constant.
Now, if there are different kinds of load, such as an inductive or capacitive load, or a mixture, you get different power dissipation, because the current is not necessarily in sync with the voltage.
But that's another story.
More: In case you're interested in what current and voltage actually are, simple household plumbing is a good analogy. Voltage is like pressure, and current is like water flow. In fact, a piece of metal wire is almost exactly like a water pipe, except it is chock full of electrons that can move freely.
They don't actually have to move very fast.
It's just that if some are put in at one end, they have to leave the other end at the "same time", because they are not really compressible.