The name of the sword Aragorn carries after it is reforged is Andúril. It was made from the broken shards of Narsil, the sword once wielded by Elendil against Sauron.
Narsil was broken when Elendil fell during the Last Alliance of Elves and Men. Isildur nevertheless used its hilt-shard to cut the One Ring from Sauron's hand. The fragments remained an heirloom of Isildur's line until they were brought to Rivendell and reforged by Elven smiths.
Aragorn renamed the restored blade Andúril, a Quenya name meaning “Flame of the West.” Its design included seven stars between a crescent Moon and a rayed Sun, symbols linked to Elendil's lineage and Númenórean heritage.
Narsil, Glamdring, and Orcrist are all different named swords. Narsil is the original broken blade; Glamdring belongs to Gandalf, while Orcrist was carried by Thorin Oakenshield.