GASPS
SOMEONE NEEDS ME! I FEEL WANTED AGAIN!
Ahem.
Japanese katanas have to be forged special because Japan is an island. Despite all of their mountainous terrain, the iron there sucks. It's all full of impurities and whatnot and if you tried to make a blade out of it, if it even remained a whole piece when you were done, it wouldn't even be good enough to pick your teeth with it.
However, Japanese bladesmiths figured out some pretty ingenious ways to make it work for them. Taking their iron, they would first build a temporary furnace called a Tatara, a clay pipe big enough to fit any iron chunks and iron sand they had along with a lot of charcoal. They would smelt the iron into one big chunk of sparkly jewel steel called tamahagane, and that would be what they used to create a katana. However, jewel steel is still impure, so they would heat it, hammer it into a bar, and fold it over, bringing it back to the forge to do it again.
Folding the steel makes pattern-welded steel, named for the squiggly pattern you'd get from folding it over so many times. Doing this would consolidate the steel, making it denser and easier to mess with. There are two kinds of steels, low-carbon, and high-carbon. Carbon is what turns iron into steel, and the more iron there is, the harder the steel is. This means that bladesmiths would take the low-carbon steel, forge it into a V-shape, place the high-carbon steel inside like a hotdog, and forge-weld the whole thing together. Once they turned that into the blade, the hard steel would become the edge and the soft steel would become the spine, making it bend easier so it wouldn't shatter.
Then again, most smiths only had the one steel they made to work with, so instead, they used differential tempering to solve their issues. Tempering is the process of slowly heating and cooling the sword so that the crystalline structure within the steel relaxes. If you don't temper after quenching the blade in water and try to use it, your blade will shatter like glass. Japanese smiths spread a slurry of clay across the back and spine of the blade to protect it in the quench following the tempering process. The spine won't cool down quickly, keeping it relatively soft, while the exposed edge becomes extremely hard and retains its edge for a very long time.
Usually, the Japanese smith would hammer in the edge before quenching it, sharpening it, tempering it, and then giving the blade one last polish to make sure it's sharp. When hammering, they keep the blade flat on the anvil and strike at an angle, pushing the metal inwards rather than pulling it out. This hardens the blade slightly and makes it easier to put a sharp edge on when the profile of the blade has already been made. They would sharpen with polishing stones (whetstones) and it would take WEEKS. Then again, if the person polishing knew what they were doing, the blade would outlast them. Otherwise, a bad polish could ruin an otherwise perfect blade.
That's just the blade portion, we haven't even gotten to the handle, the guard (tsuba), and the fancy decorations! I'd recommend watching Man at Arms on YouTube and seeing their katana builds like the one they did for Kill Bill. They have visuals to explain what they're doing. Happy writing!