J.R.R. Tolkien used Tengwar for the inscription on the One Ring.
Tengwar is a writing system created by Tolkien for the languages of Middle-earth. The Ring inscription appears in a calligraphic form of Tengwar, with the letters arranged in lines around the Ring. The script can represent several languages, so identifying the writing system is separate from identifying the language of the inscription.
The words themselves are in the Black Speech of Mordor. This distinction explains why answers such as Black Speech and Tengwar can both seem plausible in discussions of the Ring, but they answer different questions: one names the language, while the other names the script.
Tolkien invented several scripts for his legendarium. Cirth is a runic system associated especially with Dwarves, while Sarati was an earlier writing system in the fictional history of the Elves. Tengwar became the best-known of Tolkien’s scripts because it appears on the Ring and in many published illustrations.