First and foremost, the Sword of Elendil, a.k.a. Narsil, is proof of Aragorn's claim to the throne of Gondor. When he waves that broken sword around, everyone knows that he is a descendant of Elendil, the second-to-last High King of Gondor and Arnor. But Narsil also reminds us that Elendil's blood line has been separated from the High King's seat for a very long time. The fact that the sword has been left broken since Elendil’s battle with Sauron shows that things aren’t going so well for his descendants.
It's not until Aragorn reforges the Sword of Elendil into a new sword, Andúril (Flame of the West), that he begins to redeem the honor of his family line. When Aragorn remakes Andúril, he declares symbolically that he plans to return to Gondor and take his place as the long-lost heir of Elendil. And like Frodo carrying Bilbo’s sword Sting, Aragorn’s use of Andúril demonstrates that he is also taking in hand the heroic histories of his ancestors, Elendil and Isildur. The reforging of Andúril is only the first part of Aragorn's quest to restore the great kingdoms of his forefathers. But his sword repair is important symbolically, since Narsil originally broke when Elendil died in battle facing the same foe that Aragorn must now confront. We almost expect him to say, "My name is Aragorn... you killed my forefather... prepare to die." You never know.