The fictional Elvish script used for many Middle-earth languages is called Tengwar.
Tengwar is a writing system devised by Tolkien within his legendarium. The name is Quenya for “letters.” Tolkien designed it primarily through the fictional history of Fëanor, and its forms could represent different languages according to distinct modes. This allowed the same broad script to be adapted for languages such as Quenya and Sindarin.
The One Ring inscription is one of the best-known examples of Tengwar in popular culture. Although the inscription is written with Tengwar letter shapes, its language is the Black Speech of Mordor, demonstrating that the script and the language are separate things.
Cirth is another Tolkien-created writing system, associated especially with angular runes. It is therefore a common alternative answer, but Tengwar is the script most prominently recognized from the Ring inscription and Elvish writing.