Falchion- Medieval European sword with a broad blade and single edge
Cut-and-thrust sword- Renaissance sword, similar to a rapier, but with a broader bladed designed to (duh) cut and thrust, as opposed to the rapier's thrust-only design
Tulwar- Indian sword, similar to a saber or scimitar
Claymore- large, two-handed sword native to Scotland
Jian- double-edged Chinese sword, sometimes called a T'ai Chi sword. Seen in movies like "Crouching Tiger, Hidden Dragon". Usually single-handed, although a two-handed version exists
Tachi- Japanese sword, similar to a katana, although it tends to have a more pronounced curve to the blade. Used by cavalry, it actually precedes the katana in development
Cutlass- short, curved sword used in the 16-1800's on sailing ships
Estoc- a long, stabbing Medieval sword with no real edge, designed to pierce armor
Dao- Chinese broadsword, often with a single edge. There are a few variations of this.
Nodachi- large Japanese sword, similar to the Katana in design, except upwards of six feet long and used to cut down horses.
Hooked swords- an odd Chinese design, the blade is curved inward, similar to the shape of a cane. Supposedly used to hook and cut the tendons on the legs of horses.
Zweihander- literally, "Three Hander", a German/Swiss design from the Renaissance that reaches nearly six feet long
Gladius- Roman short sword
Spatha- longer Roman weapon, used primarily by cavalry
Khopesh- Egyptian design that vaguely resembles an elongated sicle
Arming sword- your standard Medieval single-handed sword
Long sword/War Sword- Medieval sword designed to be used with two hands. Often carried on the saddle of the horse as opposed to on the person.
Hand-and-a-half/Bastard sword- swords from the late Medieval period that were designed to be used with either one or two hands. Instead of having parallel edges like early Medieval swords, they tapered a bit more to a point, and sometimes had finger rings so that they could be gripped slighly above the cross-guard
Saber- curved, single-edged cavalry weapon
Wakazashi- Japanese short sword, single-handed, of similar design to the katana
Backsword- single-edged sword with a straight blade, used in Renaissance times
Rapier- Renaissance stabbing sword with a long, thin blade. Sometimes used in pairs, called a "case of rapiers"
Dha- curved, single-edged sword from Thailand, often resembling a katana
"Medieval Swordsmanship" and "Renaissance Swordsmanship" by John Clements, among others