The writing system J.R.R. Tolkien created for many Elvish languages in Middle-earth is Tengwar.
Tengwar is a constructed script whose letters represent consonants and whose vowel signs are called tehtar. Tolkien designed it as part of the internal history of Middle-earth, attributing its invention to the Elf Fëanor. It appears in inscriptions, manuscripts, and decorative writing connected with the legendarium.
The script is used for several languages, including Quenya and Sindarin, although the relationship between a script and a language is not one-to-one. The same writing system can represent different languages according to their sounds and spelling conventions.
Cirth is another Tolkien-created script, especially associated with inscriptions and the Dwarves. Sarati is an earlier script in Tolkien’s invented history, while Angerthas refers to expanded runic systems. These names are related to Tolkien’s writing experiments but are not the canonical answer here.