Music notation gives you a record of exactly how to play a piece of music. But how do you write a record of a dance?

The staff / stave notation for music dates back about ten centuries, to the work of Guido d’Arezzo (creator of the Guidonian Hand and do-re-mi). At this point it is ubiquitous and standard. Dance has no such standard notation system. It’s really difficult to record dance moves because they involve so many variables: core posture and weight distribution; limb and joint position, direction, and movement; the whole body’s own position and movement through space; and the timing of it all as it relates to music.
Despite all these challenges, many choreographers and dancers have tried to create dance notation systems anyway. The picture above is from an early 18th century book of dance that uses the Beauchamp-Feuillet notation system. Designed specifically for baroque dance, this system is extremely precise while also being completely opaque to an outsider. The semi-circles in the top row represent half-turns. On the second row, the short lines branching out of the central bar represent some fancy footwork (sinks and springs). I have no idea what the rainbows on the third row are. Spins? If you want to wade through the details yourself, the third link below contains an excruciatingly detailed summary of this notation method.
Beauchamp-Feuillet describes baroque dance well but doesn’t capture every possible kind of human movement. The solution to this dilemma: either create a notation system localised to a specific dance style, or try for the holy grail of a general dance system. Tap dance, for example, has Kahnotation (gotta keep track of all those grace taps, flea hops, heel drops, and double snaps!). Tango, Korean dance, and Morris dancing all have notation systems of their own.
This happy chap, Rudolf von Laban, was both an early pioneer of modern dance and the inventor of a general dance notation system called Labonotation:

That’s not abstract art behind him, by the way, that’s his system. Other generalised systems are less abstract, like Friedrich Zorn’s beautiful 19th century Zorn Notation:

Given that we can now record dance moves directly onto video, it seems unlikely that we’ll see any new dance notation systems in the future.
Juggling notation solves a similar, though significantly easier, problem.
http://www.juggling.org/help/siteswap/faq.html