Brightsteel was a special rare metal used to make Dragon Rider swords. It was a metal from a meteorite that crashed in Du Weldenvarden and was subsequently found by Rhunön. When she forged with it, she discovered that the metal had special properties. She scoured Du Weldenvarden for the metal, saving all the brightsteel she could for making the swords of the Dragon Riders. When she was asked to make a Sword for Eragon I, she made it out of this material.
Because of the special properties of brightsteel, Rhunön would need several weeks to forge a sword. She also did not use Magic when creating a sword, further adding to forging time. When the brightsteel was being molded, Rhunön would forge its shape to fit the Rider's fighting style, and use magic to decide it's color, usually the color of the intended rider's dragon. 
Zar'roc, Morzan's sword, was one such sword forged by brightsteel. It took on the color of Morzan's Red Dragon. The sword was designed to fit Morzan's hand perfectly for a one-handed fighting style. Murtagh took Zar'roc from Eragon during the second Battle of The Burning Plains. Coincidentally, Zar'roc's blade color matches that of Thorn's.
Brisingr is Eragon's sword. He and Saphira found some brightsteel under the Menoa Tree: the amount he brings, according to Rhunön, was enough to create several swords. Brisingr was forged as a hand-and-a-half sword and was colored a bright blue, the same shade and color as Saphira. Due to time being of the essence, Rhunön broke from her usual practice of not using magic in the creation of her swords and Brisingr was completed in less than a day. Because she had sworn to never again make a Weapon of Death, Rhunön controlled Eragon's body to forge the sword. Despite all of this, Rhunön felt that it was her best work.
Other fiction and real-life examples Edit
Legends and myths have various examples of weapons created from meteorite ore. Achilles' armor and sword were said to be made of meteorites. Norse mythology possesses numerous heroes who wielded spears and swords made from the metal of a fallen star. The Vikings were known to have created real weapons from meteorites, due to the superior strength and toughness of the nickel iron alloy present in the meteorites. Meteorite weapons were highly valued in the ancient eras as at the time meteorites were the only known source of iron, a metal that was far superior to the bronze that was more commonly used.
In the game Dragon Age: Origins, one of the best swords in the game, Starfang, is forged from metal from a falling star.