Dance notation is a visual representation of dance and other kinds of human movement used to record or document sequences of motion so that others may learn and perform them. There isn't one notation system but a few prominent systems in western cultures are Labanotation and Benesh Movement Notation.
For an overview, see the entry on Dance Notation in the The Oxford Dictionary of Dance (login via the institutional login link). If you're interested in a more detailed historical account of the development of Labanotation and Benesh movement systems, see Victoria Watts' 2015 article Benesh Movement Notation and Labanotation: From Inception to Establishment (1919–1977).
Labanotation, a system for analyzing and recording human movement, was created by Austro-Hungarian choreographer and dancer Rudolf von Laban (1879-1958). He developed the system in the 1920s. The word Labanotation comes from the joining of Laban's name with the word "notation". Labanotation continues to be favored by those working in non-classical dance, while Benesh is typically used for ballet.
The below are recommended books for acquiring certifications at the at the elementary and intermediate Labanotation theory levels from the Dance Notation Bureau (DNB). For more information, see the DNB education page.
Created by husband and wife team Rudolf and Joan Benesh in 1955. They began teaching it in 1956 at the Royal Academy of Dancing. In 1962 the Beneshs founded the Institute of Choreology, later called the Benesh Institute, which was eventually absorbed into the Royal Academy of Dancing. The Benesh Institute professionalized the work by training choreologists and working to preserve notated scores. Benesh notation is used on ballet scores around the world while Labanotation is typically used in non-classical dance.