The katana is the Japanese longsword (Daito), although many Japanese use this word generically as a catch all word for sword "katana" (pronounced [katana]) is the kunyomi (Japanese reading) of the kanji 刀; the onyomi (Chinese reading) is "tou" (pronounced [to:]). It refers to a specific type of curved, single-edged sword traditionally used by the Japanese samurai. The scabbard for a katana is referred to as a saya.
It is primarily used for slashing, and can be wielded one- or two-handed (the latter being the most common mode). It is worn with cutting-edge up. While the art of practically using the sword for its original purpose is now somewhat obsolete, kenjutsu has turned into gendai budo - modern martial arts for a modern time. The art of drawing the katana is iaido, and kendo is an art of fencing with a shinai (bamboo sword) protected by helmet and armour. Old koryu sword schools do still exist (Kashima Shinto Ryu, Kashima Shin Ryu, Katori Shinto Ryu).
Japanese swords and other edged weapons were manufactured by an elaborate method of repeatedly heating, folding and hammering the metal. This practice was originated from use of highly impure metals, stemming from the low temperature yielded in the smelting at that time and place. In order to counter this, and to homogenize the carbon content of the blades (giving some blades characteristic folding patterns), the folding was developed (for comparison see pattern welding), and found to be quite effective, though labour intensive.
The distinctive curvature of the katana is partly due to the differential heat treating it is subjected to. Unlike swords produced in many other locations, Japanese smiths did not harden the entire blade, but only the cutting edge. The hardening process will make the edge part of the blade contract less than the untreated steel when cooling down, something that aids the smith in establishing the curvature of the blade. The combination of hard edge and soft back of a katana and other Japanese blades is what cause them to be resilient and yet retain a good cutting edge.
Many myths surround Japanese swords, the most frequent being that the blades are folded an immense number of times, gaining magical properties in the meantime. Note that with each fold made by the maker, every internal layer is also folded, and so the total number of layers in a sword blade is two to the power of the number of folds made. As for magical properties, see above.