If it's the sword Newsround's website thinks it is, it's called Anduril, and was broken by Sauron thousands of years ago, just before the leader of men, Isildur, used what was left of it to cut the ring from Sauron's hand.
The pieces of the weapon - which was originally called Narsil - were shown in the first film The Fellowship of the Ring at Rivendell.
Aragorn becoming more kingly
According to the books it is put back together, or re-forged, by the elves.
Aragorn having that sword is important because it means he is beginning to live up to his role as the King.