is it like pulling hot metal taffy?
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It’s extruded. The metal is squeezed out through a hole of appropriate size.
Is that like pulling hot metal taffy? I don’t know.
Ah. I do know what you mean about the taffy. Yes, you can make wire like that too. By winding it progressively faster, you make it thinner. It’s called “drawing”.
Some specialty wires (especially heavier gages) are extruded. Most is drawn; a method somewhere between extruding (ala the Playdough fun factory) and taffy pulling.
As indicated above, drawing is the most common method locally. The local wire and wire-product (chicken wire, stuco lattice, etc.) manufacturers have big drawing machines and draw the wire in a variety of sizes. I think extrusion is done initially at metal working facilities to produce the stock for the wire drawing companies. Insulation is applied to the drawn wire through a plastic extrusion process with wire fed through the center of the die.
There’s a nice diorama at the Boston Museum of Science showing wire-making in the eighteenth century. A large size wire is pulled through a funnel-shaped hole in a plate. It enters big and comes out small. The wire is held by a pair of tongs, sort of like ice tongs, and the “handle” that holds the far end of the tongs also pushes them close together, so they maintain a tight grip on the end of the wire. The “handle” is attached to a cable on a winch. The apprentice gets to strain and grunt as he pulls on the winch, effectively compressing the metal by force alone as it goes through the die (that “funnel”) and gets reduced in diameter. I imagine that:
1.) The metal must get awfully hot from the drawing. With fine wire you probably get air-cooling fairly rapidly, but I’ll bet the apprentice has to keep dumping water on the die to cool it.
2.) The wire being drawn must break off more often than you'd like, causing heartbreak.
3.) The drawn wire has to be annealed (heated to high temperature for a period of time -- longer than it stays hot from the pulling) in order to keep it from being excessively brittle aftere being drawn.I work for the company that makes more magnet wire than anyone else but I have little to add because the question has been answered so well by others. Copper is most often used not because it’s an excellent conductor but becaus it’s extremely ductile, more so than other metals. That keeps it from breaking when drawn out so thin. We do the whole enchelada from mining or in Arizona, New Mexico and Chile to refining and making the finished wire and cable.
Copper wire of various gauges is made by finding old pennies (new copper coated zinc pennies won’t work) and then finding two lawyers to fight over the penny, the result being like two cats tearing at a wool sweater.
Really.
Q. How did Mickey and Minny Mouse meet?
A. They were just drawn together.
I’m a little confused here. Wouldn’t the sauce mess up the drawing machines? Or, are there special Taco Drawers? And, are Taco Drawers like Hot Dog Buns? 
I once did the control program for a machine that makes optical fibre.
They start of with a large piece of solid cylindrical glass, about 1m high by 10cm wide.
This is then placed in a gripper at the bottom of the machine and a gripper is clamped on to the top.
A mechanical circular furnace, round the outside of the glass is then started up and the two grippers, top and bottom, start to rotate, at the same speed and in the same direction.
Once the preform is heated to a sufficient temp the top gripper stats to rise, at a predetermined speed, thus stretching the preform to the desired length.
The burner travels up and down the preform constantly monitoring the temp and the thickness of the preform. Once the desired thickness is achieved the burner is turned off and the process is finished.