Narsil

Narsil refers to the shards of the original sword once wielded by Aragorn's ancestor Elendil. However, the great king died in battle and was used on Sauron by Isildur, who cut off the ring Sauron wore.The sword was kept in Rivendell until Elrond had the sword reforged as Andúril.

The Lord of the Rings: The Fellowship of the Ring
The sword first appears in the flashback told by Galadriel. However, it somehow landed in the possession of Elrond, the lord of Rivendell.

One day, Aragorn sits at a  nearby location to the sword where Boromir approaches. When he asks who Aragorn is, the man simply answers that he is  a friend of Gandalf the Grey. The man of Gondor then looks at the sword and picks up the blade with the hilt still in place. As he looks at it, Boromir accidentally cuts his finger and realizes its still sharp. As Aragorn eyes him,Boromir looks away from teh sword and claims it's nothing more than a broken sword and throws the hilt onto the ground. Aragorn goes over and places it on with the rest of the shards before engaging a conversation with Lady Arwen.

Return of the King
The sword is reforged as Anduril and given to Aragorn by Elrond.

Other appearances
Narsil appears in The Hobbit: An Unexpected Journey where Bilbo is wandering around Rivendell and spots the swords' remains.

Difference from source material

 * Narsil was reforged as the sword in The Fellowship of the Ring and wielded by Aragorn before he left for the quest to Mordor.
 * However it is reforged in Return of the King and given to Aragorn by Elrond.