You've probably seen multiple instances of this in your daily experience. A fuse is a thin conductor between thick electrodes. A light bulb is a very fine conductor between thick electrodes, encapsulated in a vacuum bulb. If you use tungsten electrodes, you can easily melt copper in the manner I've described -- that's how TIG welders work.

Responding to your question about electric current:

Quite simply, use leverage! Take a transformer with multiple (N) windings on the primary and a single winding on the secondary. Putting one (DC) amp through the primary will induce N amps through the secondary. With induction, you can use a low current to induce high current. Or correspondingly, transform low (AC) voltage to high voltage -- this is how high voltage power lines work.