There are lots of different types from lots of different cultures - calligraphy, since the definition is 'the art of beautiful writing' has a next to infinite number of forms - in lots of different languages and countries throughout the world. Some examples from the Western world are - Gothic Script, Georgian, Latin Script etc. Its important to also consider Chinese calligraphy, Katakana (Japanese Calligraphy), Indian and Arabic calligraphy forms. I hope this answered your question.
According to research, China alone has four different styles of calligraphy. These are official script, cursive script, standard script, and semi-cursive script. These four styles can be applied to western calligraphy as well. Even before the advent of computers, there were many hundreds of styles of calligraphy. Calligraphy used to be how the written word was copied onto a page, before the advent of the printing press, which uses a typeface.