The fictional ring that makes its wearer invisible in J.R.R. Tolkien’s stories is the One Ring.
The One Ring was forged by Sauron in the fires of Mount Doom during the Second Age. Sauron placed much of his power into it so that he could dominate the other Rings of Power and control their bearers. Its destruction is the central objective of The Lord of the Rings.
The ring can make ordinary mortals difficult to see, but invisibility is only one of its effects. It also exerts a powerful influence over its owners and can reveal or attract dangerous forces. Bilbo first finds it in The Hobbit, where it is not yet presented with the full history later revealed in the sequel.
Other rings in Tolkien’s legendarium include the Elven rings Narya, Nenya, and Vilya. The Ring of Barahir is a different object entirely, so neither name is a correct substitute for the One Ring.