The fictional writing system J.R.R. Tolkien devised for many Elvish texts was Tengwar.
Tengwar is a constructed script attributed within Tolkien's legendarium to Fëanor. It was designed to represent the sounds of languages such as Quenya and Sindarin, although it could also be adapted for other languages. Its letter forms are recognisable from the inscriptions and title pages associated with Tolkien's works.
The One Ring inscription is written using Tengwar characters, but its language is the Black Speech of Mordor. That example illustrates why script and language should not be treated as the same thing. Tengwar is the writing system; the words belong to a specific language.
Tolkien also devised Cirth, a runic script associated especially with Dwarves and used for inscriptions. Sarati is an earlier writing system in his invented history, so it is not the canonical answer for the best-known Elvish script.