Plate armor was produced assembly line style. A smith would first purchase iron ore and heat in a blast furnace, which would cause the impurities to liquefy and drain away. This would produce a porous "bloom" of iron. While the bloom was still soft from the heat, it would be hammered to knock out any remaining slag and then hammered into bars. The smith would either sell these iron ingots or carburise them to make steel. The armorer would purchase these ingots, and after softening them with heat use a hammer to shape them into the plates of the armor. At this point the individual plates of the armor would be fully formed but with a black and rough finish. The armorer would pass the plates on to polishers, who would polish them to the desired finish. The polishers would make sure all of the pieces fit together correctly and then hand them off to leather-workers. The leather-workers would attach the leather straps and buckles that attached the plates together to form the elements of the harness (suit of armor). If the harness had been commissioned, it would then be shipped to the customer. Otherwise, it would be put up for sale. Individual pieces were sold as well.

That's a very basic run down and there's many variations to the process in different regions and time periods. Commissioned armor would have the extra step of carefully measuring the customer to produce a perfectly fitted harness, similar to having a modern suit tailored versus buying off the rack. Sometimes the polishing step was skipped and the armor was left black and rough from the hammer. Additional decorations were often added. Armor could be painted, engraved, filed, fluted, covered with fabric, or later in the period acid etched. Punch-and-file work and fluting were done by the armorer, the rest would have been done by special artisans.

If you want to learn more, you can start with reading armorer Craig Johnson's 1999 essay on the subject: http://www.oakeshott.org/metal.html

If you're really interested in the subject the best source is Dr. Alan Williams' seminal and comprehensive work, the 1000 page The Knight and the Blast Furnace. Since it's essentially a textbook it's fairly expensive, so you'll probably want to find it in a library or download the ebook from scribd.