These are some different forms of musical notation:
Hand Signs
Letter notation
Solfa/sargam
Neumes
Tablature
TUBS (Time unit box system)
Stave notation
Graphic notation
Notation can have different functions:
to preserve music
to help remember music, an aide memoire
to keep a record of what you have composed
to assist in refining or revising your work
to standardise performance
to communicate musical ideas to others
to develop a common cross-cultural (i.e.non-verbal) language
to provide a visual reminder of the sounds of the music
to promote the commercial aspects of the music
to extend the range of performance situations, or develop a wider audience
to enable composers to organise and manipulate more complex musical ideas
to help people to learn music
to support study and analysis
Some forms of notation are better at some of these functions than others.