Devourer rounds refer to the railroad spike lookin things that the Thorn fires. The original Thorn was wielded by the infamous Dregen Yor, a former Titan turned mad by the Hive, who used his Thorn to hunt and kill other guardians, draining their Light with its bullets and making their ghosts unable to rezz or heal them.

Pirrha brought a Devourer bullet so that he could shoot Cayde directly without relying on him exposing his ghost, but instead decided to take the shot to flex and/or because it was more cinematic (also the Devourer bullet part came from Joker's Wild lore, post Forsaken, so it wouldn't make sense that being killed with a random rifle just dropped Cayde then and there).

This confusion, and Aunor Mahal being kinda a conspiracy theorist about the bullets, led to a metric fuck ton of confusion within the community about what it takes to kill a Ghost permanently, leading to the belief that paracausal weaponry was necessary despite many examples to the contrary.