In addition to the great information on traditional Japanese swordcrafting given in other answers, I’d like to just mention that most modern katanas, purchased from reputable manufacturers and intended for actual use, are made of high quality modern steel alloys in factories or workshops. Many aren’t even differentially tempered (harder on the edge than the spine). The reason for this is simple: steel is probably now the least expensive part of swordmaking. The labor of folding a blade to purify it, joining a hard carbon steel edge to a softer blade body, etc., just isn’t necessary to make a good sword in modern times.

You might not consider it a real katana if it’s not made in the traditional fashion, but you should know that you can get something that looks identical and performs just as well (and may be more durable) for much, much less cash than a katana made to historical specs by a master smith. I’m glad the craft still exists, though.