Oh hey, a ballet question. Great!

Off the bat, for a long time ballets were passed down through much less "official" means than sheet music, in the sense that their reproduction depended on consulting people who were in previous productions or deciphering notes from original choreographers. This was still the case in the early Soviet Union (my research in particular touches on this), where reproductions of classical ballets relied heavily on the expertise of long time members and leaders int he ballet. In fact, this is probably still the case in practice many places today, even if institutions have become a bit better at preserving choreography. Individual style of notation helped to preserve some choreography, but you're quite right that lack of a universal standard like musical notation made it difficult if not impossible to precisely preserve choreography. When most companies to reproductions and reconstructions of ballets, the choreography changes some. Exact reproductions are large undertakings and are actually not too frequent.

Standardized styles of dance notation did, however, get produced. I'm just going to talk about one right now - Labanotation. Labanotation is a form of movement notation that is capable of very precisely capturing movement on paper. It's very complicated and even as someone who studies dance I have to admit that I need to struggle my way through it with a key when I have occasion to need to read it, and writing it from scratch on any kind of large scale (like a whole ballet) is basically out of the question except for specialized experts (which did and still do exist, although there are quite few). Labanotation comes out of the work of Rudolf Laban, who was interested in dance theory and who came up with this system of codifying the notation of movement in the interest of better being able to theorize about dance and movement (he is also well known for Laban Movement Analysis). His work came mainly out of the first half of the twentieth century, but I think it wasn't until about 1960 (give or take a few years, I honestly can't remember) that Labanotation became offically codified as a single set of symbols and "rules."

Labanotation looks like this:

https://s-media-cache-ak0.pinimg.com/originals/95/42/0e/95420e9187bc2db0eb6a7a59a3263c2b.jpg

As you can see it's entirely inscrutable without being taught how to read it. Music notation is too, of course, but the complexity and range of possible movement obviously makes notating movement incredibly complex. The short version is that it works a little like a musical staff except that it is read bottom to top (and left to right in terms of columns). Movement for any given body part can be recorded, as can quality of movement.

Labanotation was particularly useful in the mid-twentieth century as choreographers became more interested in being able to officially copyright their work. Being able to precisely record on paper a dance made it much easier to claim copyright. It wasn’t until the late 1970s that choregraphy become copyrightable in the United States. (Here is some more information on choreographic copyright in the US: https://web.csulb.edu/~jvancamp/copyrigh.html) However, I can’t say that it has ever been widely used. The notation’s complexity (again, a necessary feature) makes it a huge undertaking to learn and write and read. Although many dancers are taught a small bit about Labanotation as part of their training, expertise is rare and really requires studying it specifically at length. So while this does exist as a way of recording dance, it is not a widespread practice.

George Balanchine was one of the most influential dancers and choreographers in the 20th Century. There now exists “The George Balanchine Trust” (www.balanchine.com) which is charged with maintaining the integrity of his work through copyright of his works and officially licensing out his choreography. They have very high standards for the companies they allow to perform the work and so forth. But the the point here is that this is what it looks like in practice when copyright and precise preservation of ballets is taken seriously. It takes a lot of effort, time and money.

In practice, ballets change over time. Whether it be at the New York City Ballet or the Bolshoi Ballet or any number of other professional ballet companies in the world, ballets tend to be made, performed, and then later revived. If you go see Swan Lake tonight, it’s going to be different than the one performed 30 years ago, even if they have a lot in common. The degree to which the original choreography is maintained is more a function of who is leading the revival, what dancers the choreographer has to work with in the new version, and what knowledge of the original choreography still remains in the institution (by whatever means). Video plays a very large role in choreography these days in practice. Although videos of dances are not as useful for issues of copyright, they are enormously useful when choreographing. Still, we still lack today the kind of official, universal and, perhaps as we’ve seen, widely practiced style of notation that exists in music.

I know of three systems used in the French Baroque: Favier's, Lorin's, and Feuillet's. You can read a little bit about them here.

You can see some influence of musical notation in those systems (in many cases you see music notation next to dance notation, to try to give people an idea of how to organize things). Dance was influential in music during the Baroque, not just because both were practiced together, but because it gave musicians patterns for composition and performance. These notation systems are of interest to musicologists because they give us some information about what people were doing. These systems are obviously not as developed as Labanotation (already mentioned by /u/TenMinuteHistory), and I understand they are kind of dead these days.

You are probably interested in looking at the Stepanov choreographic notation. Vladimir Stepanov created a system in the late 19th century (it is based on music notation). I understand this notation was actually used to communicate coreography and to mount ballets in different cities, and that some of Marius Petipa's choreographies were encoded with it. The Harvard University Library Theatre Collection has a collection of these coreographies.

Benesh movement notation is another system (invented in the 1940s), the Royal Academy of Dance has a collection of coreographies in this notation.

The Eshkol-Wachman movement notation is a system to record movement on paper or using a computer. It is not specific for dance, and has been used for other things (like studying animal behaviour).

Search for "choreology," (the study of dance notation, and/or the recording of dance movement by notation). I've seen the Spanish version of this word (coreología) sometimes used to name the study of dance.