The inn visited by Frodo and his companions in Bree is called The Prancing Pony.
The Prancing Pony stands in the village of Bree, a settlement where Men and Hobbits live near the eastern edge of Eriador. Its innkeeper is Barliman Butterbur, who becomes important because a delayed message from Gandalf reaches Frodo through him. The inn is also where Frodo first meets the mysterious ranger Strider, later revealed as Aragorn.
The hobbits stay at the inn after leaving Hobbiton and before beginning the dangerous journey toward Rivendell. Their presence attracts unwanted attention, and the building becomes a key early location in The Fellowship of the Ring. The inn’s common room also illustrates Tolkien’s mixture of ordinary rural life with the growing threat beyond the Shire.
The Green Dragon is a different inn, located in Hobbiton. It is associated with Hobbiton drinking culture and appears in the wider Shire setting, but it is not the Bree inn where Frodo meets Strider.