I always like to give the water and plumbing analogy here.
If you think of the voltage as the diameter of a pipe, and the current as force of the pump moving the water. The water coming out of the end of the pipe, say in gallons per minute, is the power.
Now, if you increase the diameter of the pipe (the voltage), you can get more water through at once - thus increasing the "power".
Also, if you increase the force of the pump (the current), you force the water through faster. This however increases the pressure inside the pipe, so you need to use a pipe with a thicker wall. This is equivalent to the size of the conductor. Bigger conductors (or thicker walled pipes) can hold more current without "bursting" (melting).
Of course, making the pipe too big (too high a voltage) and the eco-protestors all turn up and start complaining about the damage laying the pipe does to the countryside, and people start to get hurt (electrocution), so you need to build bigger and better barriers to keep people away (insulation).
So, you can see that there is always a balancing act between the thickness of the pipe wall and the diameter of the pipe in order to get the amount of water through it that you need.
Yes, I know the analogy isn't quite 100% right, but it's good for getting an understanding of the relationships between them.