Hilt-and-point, or linked, sword dances are a style of dance which originated in Germany (probably in Nuremberg) in the middle of the 14th century. From there, they spread all over central and western Europe; evidence of the tradition has been found by folklorists in England, Scotland, Spain, France, Italy, Germany, the Low Countries, Romania, Austria, and Scandinavia. A particularly showy variety of these dances known as "rapper dances" began in England in the 19th century, peaked in the period between the World Wars, and is still going strong today; many other places have either preserved or revived their sword dance traditions as well. In dancing these dances today, performers continue a custom which connects them to the medieval bourgeoisie and so to over six centuries of history.
The concept of using weapons in a choreographed dance can be definitively traced back as far as ancient Greece, but there's no reason to believe it isn't as old as the human tendency to attach significance to implements of violence. The vast majority of these, however, have been what 19th-century folklorists dubbed "Pyrrhic" sword dances, in which the dancers enact mock combats in time to the music. Such dances appear in many cultures all over the world, and the combative aspect may be highly stylized, while some are very nearly as realistic as the fights in the average Errol Flynn film. (The fencers reading this will have just sighed. Mr. Flynn's fights aren't very close at all to reality. But they have an enchanting quality to them which, in the context of a dance that does not even pretend to be an actual combat, is well worth seeing.)
The dance originated by the Nuremberg Cutlers' Guild in the years following the Black Death, however, is something else entirely. In the linked sword dance tradition, the dancers each hold a blunted sword by its hilt in their right hand, and spend some or all of the dance forming themselves into a chain by grasping another sword's point in their left hands. A variety of figures of varying degrees of intricacy are then formed and unformed, all while staying in a loop of swords. Along with these linked figures, it appears that the early German sword dances also included pyrrhic elements; in some descendant traditions, these retain varying levels of prominence, while in others (such as the northern English sword dances) they have been eliminated entirely. There is some reason to believe the pyrrhic elements were most prominent in Spain and Italy, while Germany and the Low Countries emphasized the linked figures. Certainly this is the case when we compare modern Basque espatadantza with the always-linked visual representations that occur in Dutch and Germanic visual sources from earlier times; projecting these aspects of the modern southern traditions backwards, however, is supported only minimally by the historical evidence.
To the best of my knowledge, no work has been done on the antecedents of the first linked sword dance (but if anybody knows of any, I'm intensely curious!), but it's quite likely that in taking up swords to perform a flashy, showy dance, the Cutlers were engaging in an early community relations event, helping to enliven a local civic festival while creating good will for their guild. Soon other guilds in other towns took up the responsibility of performing such dances for their own populace, and the style spread throughout the Holy Roman Empire over the following centuries. In time the association with trade guilds was lost; Olaus Magnus, for instance, describes what is unmistakeably a linked sword dance performed in Scandinavia in his History of the Northern Peoples (1555), saying of its performers only that they are young men.
It is unclear when sword dances began to be performed in the British Isles. This is in part because they were often associated with performance dances done in disguise, a tradition which gained prominence no later than 1450, but references to masked dancing often exclude any mention of what if any props the dancers are carrying until the 17th century. It's possible that England adopted some form of sword dancing in this vein as early as the 15th century, but it is also quite possible that they were not introduced until Dutch coal miners began to immigrate to England in the late 16th century or even later, bringing with them a distinctly Germanic style of sword dancing. (It's also possible both happened; this would explain the disparity between morris dancing, which resembles a highly-stylized pyrrhic dance in which the swords have been replaced by sticks, and the sword dances of the coal-rich regions of the Tyneside, which are linked from start to finish. The notion that sword and morris dances are closely related or even part of the same tradition, however, clearly goes back as far as both styles have existed in Britain.)
Modern English sword dances are divided into two families, first recognized by Cecil J. Sharp in the early 20th century. The older style, which Sharp imagined to have been prehistoric and Pagan in origin (unaware of the historical record which makes it clear they began in an already entirely Christian late medieval Germany), are the ones he terms "longsword dances," a name still in common use among sword dancers in the English tradition today. In these dances, well exemplified by the Kirkby Malzeard dance (which greatly resembles Olaus' description of a Swedish dance), the dance is stately and elegant, and has lost much of the acrobatic drama seen in early visual sources. The dancers carry swords which resemble the genuine article, though they are sometimes a bit more flexible to facilitate those passages in which the swords are woven into a star or other shape and displayed to the audience. Typically, the dancers are led by a Captain, whose part in the actual dance is minimal; he usually introduces the dancers by singing a particular traditional song, and may be the subject of a mock execution partway through the performance. It is my experience that these dances can be enjoyable to dance, but are not especially exciting for an audience.
The newer type are the "short sword" or "rapper" dances, so termed because the "swords" (which, with handles at both ends and likely originating from a now little-used horse grooming tool, have almost no resemblance to actual weaponry) are about six inches shorter than those used in English longsword. (The origin of the word "rapper" in this context is unkown; it may or may not be related to "rapier.") In these dances, the swords are quite flexible, able to be curved even into a complete circle; the dancers (who are always five in number, often with the addition of a Captain and a man in drag called the "Betty" or similar names) are thus able to perform figures which would be impossible in a longsword dance. These dances are quite showy, and are always choreographed with the entertainment of the audience in mind; since the early twentieth century they have often been performed competitively. The Newcastle Kingsmen, a consistently high-performing team since their establishment shortly after the Second World War, exemplify the modern rapper tradition. As seen in their performance, these dances often involve stepping derived from the Irish step dance tradition; this, like the rappers themselves, is likely an innovation of the nineteenth century, when it came to be the Irish rather than the Dutch who most frequently immigrated to northern England to work the coal fields. (English sword dancing has long been strongly associated with coal mining.) While these bear little resemblance to the first sword dances, the constant evolution since the fourteenth-century German origin means that the Kingsmen and other rapper teams are part of a tradition of choreographers as well as performers who have been refining their art for centuries.
In Germany and the Low Countries, meanwhile, sword dancing either died out or came very near to it, prior to modern folklorists' study of the form. However, in modern times reconstructed dances based on early visual sources have been resurrected, and are often performed in sixteenth-century costume. The Traunstein dance is not an atypical example. In Traunstein and a few other towns, the dance has once again become an annual tradition, whose performances hearken back to the sword dance's late medieval and early modern origins. While of course the details vary throughout the modern sword-dancing world, these dances provide an iconic example of how modern practice of folk customs can provide a link to the past for the participants, both performers and audiences alike.