"Regalia" is a general term for the special items or decorations that are used as marks of distinction in ceremonial occasions. When Queen Elizabeth II of the United Kingdom wears her crown and the medals of the British heraldic orders and carries the scepter, the crown, medals and scepter are regalia.
In "What You Pawn I Will Redeem," the regalia spotted by the narrator was the long-ago stolen ceremonial attire that had been worn by his grandmother during ceremonial dances at powwows. "it had all the same color feathers and beads that my family sewed into our powwow regalia."
Regalia is important because it symbolizes the significance of the person wearing it. The colors of the feathers and beads used in the grandmother's ceremonial dress probably had meaning beyond the one hidden yellow bead that allowed him to prove that it was his family's stolen property.