"Katana" (pronounced KAH TAH NAH) is the kun-yomi (Japanese reading) of the kanji 刀; the on-yomi (Chinese reading) is "to" (pronounced (TOH). It refers to a specific type of curved, single-edged sword traditionally used by the Japanese samurai.
It is primarily used for slashing, and can be wielded one- or two-handed (the second being the most common mode). The art of drawing the katana is iaido, and other martial arts related to the katana are kendo and kenjutsu.
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, 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, forcing the blade to curve. 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 the number of layers in a sword blade is the number of folds to the power of two. When speaking of folds, the layers and actual foldings are often confused. As for magical properties, see above.