J.R.R. Tolkien named the reforged sword Narsil Andúril.
Narsil was the sword of Elendil, the founder and first High King of Gondor and Arnor. In the Second Age, Elendil used it against Sauron, but the blade broke during the final battle of the War of the Last Alliance.
After Elendil’s death, his son Isildur took the broken hilt-shard and used it to cut the One Ring from Sauron’s hand. The fragments were preserved as an heirloom of Isildur’s descendants.
In The Lord of the Rings, the sword is reforged for Aragorn in Rivendell and receives the name Andúril, meaning “Flame of the West.” It symbolizes Aragorn’s royal lineage and his claim to the thrones of Gondor and Arnor. Glamdring and Orcrist are different swords from The Hobbit.