When Brom turned ten, he was presented to the eggs that the dragons had given to the Riders. A sapphire blue egg hatched for him and her name was Saphira.
Saphira I trained along with Morzan's dragon, both being taught by Glaedr and Oromis. She and Brom bonded during those years of training and became each other's only family, as Brom never saw his family.
Saphira fought during the Dragon War. Initially, Oromis and Glaedr took their students to the safety of Du Weldenvarden, but Brom and Saphira elected to join the fight. Unfortunately, she was slain by the Forsworn, almost certainly by Morzan and his dragon, while protecting Brom in Doru Araeba, causing him much grief. To avenge her death, Brom killed several of the Forsworn including Morzan and caused the death of five other members. After he slew Morzan he took his sword, Zar'roc, likely because his sword, Undbitr, was lost when Saphira died. It is highly possible that Galbatorix extracted her Eldunarí before destroying her body.
She was Saphira II's namesake. While looking for a name for her, Eragon asked Brom for a list of dragon names. Eragon then gave the list of dragon names to his yet-to-be-named dragon. He read them out, and she chose "Saphira".