I was thinking amperage was some ethereal secondary property of the electricity itself, somehow altering how "powerful" the voltage was.

Well, this is almost an entirely differently topic, but as Peter said above, both the voltage and current determine the power the electricity delivers, where P = V * I. Power can be thought of as the ability to do work, but explaining that is a whole other discussion.

So, yes, they are both properties of the electricity, which effect the power the electricity (not voltage) delivers.

I guess I'm still not getting it. Whether the water is coming from a hole in the side of a dyke along the river, or whether it's coming from a same-sized hole in the side of a fire hydrant, the amount of flow is still determined by the pressure behind that hole. How can there be two different kinds of pressure? There's either X pressure behind that hole or there isn't. I don't see how the water can represent both volts and amps.

You've held the size of hole constant (R). The water doesn't represent volts or amps. It represents the electricity itself.

Remember, measurement of pressure = volts, measurement of flow = current.

John