You may have to refine your question, as "wire" can mean anything from fine enough ( .010" or below) to be twisted strands of wire in a small flexible insulated wire in a radio to something .5" or bigger that's being drawn, pointed, cut and headed to make copper spikes and rivets.

However, if you mean something plain copper, about the diameter of the wires connecting the mouse on your computer or finer, the basic device was a machine that would continuously draw the wire back and forth through a die plate that had large-to-small holes. The wire would be drawn through the plate and around a drum, passed back through the plate to another drum, and back through the plate again. The drums would get bigger and bigger ( as the wire would get longer) and the wire and the drawplate would sit in a batch of oil ( I think lard oil was very common) . The drawplate in the 19th c. would been high-carbon or tool steel, but circa 1910 they may have begun using carbide.

That 's the basic process. There would be other things to be managed, however. The wire rod would have to be pointed, tapered, so it could be drawn through the first hole. If there was too much reduction attempted at any stage, the wire would break. There was also the work hardening of the material. Copper is pretty malleable stuff, but it would get harder at each step, and if it became brittle enough it would crack and break, and so would first have to be annealed. But annealed soft copper might not be wanted as the end product: certainly, spring-hard brass or bronze wire would be useful. So, the amount of reduction at each stage, and when or if annealing was needed , would be calculated. That was a very critical part of the process. The wire would also have to be drawn smoothly: if it stopped or jerked in the drawing, there would be a slight change in diameter and hardness, and that would create a weak spot in the wire.

There would be different machines needed for big stuff, which would have to be reduced between rollers. A rolling mill typically ran pretty fast, often with very hot metal. If that hot wire encountered an accidental obstacle anywhere in the train of rolls, it could dive off into the shop, onto the floor, into a worker. Even by 1910, you would expect guards and safety cages that would provide some safety, but you can imagine how fun it might be to remove a tangle of .5" wire from in and around a big machine, with gears, pulleys, etc. and do it quickly, in order to get the line running again.

Lastly, there would be oval , half-round, octagonal , hex, and square and other profiles of wire used. The fine versions would be found in the jewelry business. You would expect mostly metals like silver, gold, brass, and platinum, but it's possible copper would be used, as well. For that there sometimes weren't needed very, very long lengths , and perhaps special shapes, and for those the older wire drawing bench would still be used- often by the jeweler herself.

Dunell H.( 1925) British Wire-drawing and Wire-working Machinery . Constable and Company Ltd., London. https://archive.org/details/dli.ernet.19466/page/n5/mode/2up