There are a great many different types of swords, having been produced by just about every culture with metal smelting technology. Some would even say that the Aztecs
has a sort of a sword with shards of obsidian
up the edge of a wooden spine, though I would class it as a specialist club
The earliest discovered swords are short, broad stabbing blades made of bronze which were found in Turkey in 2003.
With the spread of steel technology, the additional strength offered by the material offered more options for the smith and the combatant.
The paths of development followed by the sword in different cultures was determined by available resources, technological ability and social forces.
In Europe, there were two main ancient thrusts (forgive the pun) of sword develoment, the "classical" and northern. The Romans adopted some ideas from the Greeks but modified them very quickly to their own system of warfare. The standared military gladius was a short thrusting and cutting weapon designed for formation fighting and made by government contract. The shape of this weapon shifted over time but can still be considered a distinct type. Their cavalry used a longer cutting sword. This thread of development died with the fall of the Empire and the northern thread dominated Western and Central Eurpoe for the next 1600 years, though in unrecognisably modified form.
While this may be too much of a generalisation, the northern tradition was far more individualistic. Swords were a weapon of the wealthy (it takes far more steel and work to make a sword and a spear) and were intended for single combat and as a backup in battle. There were made individually and in Germanic cultures there was a distinctly mystical tradition attached to smithcraft.
These weapons were longer typically it seems used in a more cut oriented system, though primary evidence is thin on the ground.
These weapons would devlop into the so-called migration era swords. These were straight double edged swords with wide fullers (grooves made to lighten the weapon while leaving strength, an engineering principle still used in the construction I-beam, nothing to do with blood flow), short guards and trilobed pommels. They were not particularly sharp pointed at this stage.
This would grow into the cruciform "arming sword" the single handed weapon erroneously termed a "broadsword" by Victorian museum curators, a flaw that exists to this day. These were not the heavy crude instruments of popular imagination but rather often weighed less that one pound per foot blade length used in a number of highly advanced martial arts systems. the only art whose instructional texts survive from this period is one for arming sword and buckler from about 1300.
With advanced in armour technology over this period, plate came to the fore and swordsmiths made the next jump in the armour/weapon arms race. This was the longsword/bastard sword/hand and a half sword. Typically these weighed the same or fractionally less than the katana (per unit blade length), no matter what anyone may tell you because the fuller and back edge took some metal off the blade thus lightening it.
To gain the control necessary for finding gaps in armour two hand began to be used on the weapon and the shield was used with decreasing frequency. These weapons were either used with two hands on the hilt, with one had on the weapon and the other used in controling the opponent with the integrated unarmed combat system taught with weapon combat or with one hand halfway up the weapon, turning it into a short spear (when the hands were armoured).
The single handed sword continued as a weapon in the civilian environment and here the next major developments would happen in Spain and Italy. Here the arming sword developend into the 'espada robera' (sword of the robe, Sapin) and 'spada di lato' (sidesword, Italy) as a civilian duelling weapon and these show the beginning of the comound hilts that would grow into the ornate hilts of the rapier.
The rapier developed from these weapons as a weapon optimised for the duel and not particularly well suited for street defence or battle, for which reason it was roundly slagged off by masters teaching older, more broadly applicable systems and weapons. Simultaneous with this was the development of the renaissance two handed sword, a vast weapon of 60 to 70 inches in length most famously carried by German Landknecht mercenaries to destroy pike formations. These were the heaviest combat swords to come out of europe, some weighing as much as eight or nine pounds. (Some ceremonial bearing swords were heavier)
Single handed swords for military used were following a different path of development with cages being placed around the hand to protect it, creating the broadsword proper. This was used from the 1500s to the end of the 19th century.
The rapier would eventually become so cumbersome (Joseph Swetnam taught the use of one some four feet long) that the pendulum would swing the other way, towards a shorter, lighter sword and neglecting the cut completely. This was the smallsword, a French development in. This sword had no place ouside of the duelling field.
This would eventually lighten even further to the duelling sword (epee du combat) and then the modern sport epee.
Over this period a variety of military swords were used. The saber was developed from Hungarian traditional swords and central European messer (basically machetes). The broadsword was used in some units, particularly Scottish regiments. Other weapons such as the spadroon and palache were also in use.
The development of the sword in Japan was guided by the realive paucity of resources there, particularly steel. when compared to the high volume, high quality iron mines of Northern Europe, Japan was in a problematic situation. Tha vast majority of iron available to colonists from China and Korea was alluvial ore with high sulphur content. This requires a great deal of work to become a useable sword and the swordsmiths of Japan thus developed there art to the highest level to do so.
The first swords used in Japan seem to have been straight, double edged weapons as used by the Chinese and Koreans. Development of cavalry as the most high profile component of their forces (early samurai prized skill with horse and bow well above skill with the sword) led to the development of a cavalry oriented sword, the tachi.This was a long heavy sword slung edge down from the waist. During the sengoku jidai however a significant change occurred. It has been thorised that horse casualties were greater than breeders could keep up with, but no matter what the cause, infantry became more important. This may be connected with the develoment of the katana over about 1450 to 1550 if I recall correctly. This was a shorter, straighter infantry sword used with two hands. Optimised for the cut it could also thrust, though not as well.
The katana does not have a "scalpel" or "razor" edge as some would claim (this would be far too fragile for combat) but rather is fractionaly convex, something like a smooth clamshell intended for cutting though soft and semi soft targets. The lack of high grade steel made the development of plate armour impossible without foreign trade (and eventually imported breastplates were incorporated after trade with the Portugese).
These were very sharp weapon but suffered from a certain brittleness despite the best efforts of the smiths and edge on edge contact was avoided where possible by combatants to avoid the possibility of breakage.
A great deal of nonsense has been written about these swords they could not cut though other swords or gun barrells but were simply an exceptional example of the smith's art to make an effective weapon suited to the environment with the resources at hand, like any weapon.
I'm not as familiar with the weapons of China, Central Asia, India and the Middle East/Africa.