The blade Undbitr was one of the last seven swords to be forged by the elf-smith Rhunön before the Rider War (others of these seven included Zar'roc). Made out of a meteoric ore called brightsteel, it took Rhunön twenty-four years to find the last deposit and forge these swords.
Undbitr was a different shade of blue than Eragon's Rider sword Brisingr as Brom's dragon was a different shade of blue, according to Rhunön when she first met Eragon's Saphira and inspected her scales close up. She stated that her shade of blue would have made a magnificent color for a blade, thus indicating that Undbitr was different.
After Brom had lost Undbitr, he had asked Rhunön to forge him a new sword. However, as she had already taken her oath not to forge any more weapons, she refused. Brom was so greatly angered that Oromis had to subdue him.
Since Murtagh was now using his father Morzan's sword, Zar'roc, Eragon thought about finding Undbitr and using it after finding out Brom was his actual father. Rhunön stated that, even if Eragon found the sword, it would still take him time to retrieve it before being forced to fight his enemies. It is quite possible that the sword was in the possession of Galbatorix.
The name, translated literally from Old Norse means "Biting Wound" or "Wound Biter."