In the television adaptation of The Vampire Diaries, the vampires Stefan (Paul Wesley) and Damon Salvatore (Ian Somerhalder) do not die when their sire, Katherine Pierce (Nina Dobrev), is killed. The series establishes the concepts of a sireline, or metaphorical vampire ancestry, as well as a vulnerability where all vampires of a sireline die when the sireline originator is killed. Katherine Pierce loses her vampire status and is later killed, but Damon, Stefan, and the vampires they sired, including Caroline Forbes (Candice King), are unaffected and able to continue their roles in The Vampire Diaries and its spinoffs.
When Finn Mikaelson (Caspar Zafer) is killed in The Vampire Diaries season 3, episode 18, "The Murder of One," every vampire sired by him or part of the sireline he started died as a result, leading the characters to conclude that the death of a vampire kills all the vampires they sired. However, Katherine becomes human after inadvertently taking the Cure for immortality in The Vampire Diaries season 4 finale, "Graduation." Her body is killed in The Vampire Diaries season 5, episode 11, "500 Years of Solitude" and her spirit is expelled from the mortal plane in The Vampire Diaries season 5 episode 15, "Gone Girl," but none of these events harm the vampires she made.
The explanation for this is that it was later realized that it's an Original vampire who has to die for the sireline to be wiped out, not just any vampire. In The Vampire Diaries universe, a vampire’s sire and the originator of their bloodline, whichever Mikaelson family Original vampire they are descended from, are not always the same person. The death of an Original vampire is what triggers the death of everyone in a sireline, not merely the death of an immediate sire or someone else farther up the vampire family tree. Katherine Pierce, while quite old and sired by an Original vampire, is not an Original herself, and thus her vampire descendants are at no risk from her demise.
How The Original Spell & Sirelines Work In Vampire Diaries
The spell that created the Original vampires in The Vampire Diaries, the Mikaelson family, included a metaphysical link that made their vampire descendants dependent on them for survival. Esther Mikaelson (Alice Evans), the witch who performed the spell, did this because the Original vampires were her children and she wanted to protect them in perpetuity from being targeted later by other vampires, foreshadowing the Mikaelson family motto, "Always and forever." Her husband Mikael (Sebastian Roché) wanted to make their children stronger than werewolves after one of their children was killed by a werewolf, so the entire spell was an effort to prevent any more of their children from dying.
Thus, their children became the Original vampires through this immortality spell. When a new vampire is made in The Vampire Diaries, the link to the sireline is shared with the new vampire, making them also dependent on the Original vampire at the head of the line for continued connection to the immortality spell. Interestingly, even if Katherine had been an Original, it is unclear as to whether a sireline headed by her would have died out when she became human again or become an unlinked sireline, as Klaus Mikaelson’s sireline became when Davina (Danielle Campbell) cast the Sireline Unlinking Spell on his sireline.
Another detail adding to the confusion around sirelines is the sire bond, seen with Legacies’ Lizzie and Hope. It is a separate metaphysical bond that can exist between a vampire and their sired offspring in The Vampire Diaries universe. Sire-bonded vampires obey their sire’s commands and are incapable of hurting their sire unless they can break the bond, but the sire bond and the sireline vulnerability are unrelated. Regardless, Damon, Stefan, and Caroline were not dependent on Katherine for their own survival, and since the originator of their sireline was Klaus, they also became part of the only unlinked Original sireline.