A Sireline is a figurative bloodline of vampires or hybrids and the chain of those by whom they have been sired. If the originator of a bloodline is permanently killed with a White Oak Stake, or by other means, everyone in their bloodline dies within a few hours. It was a powerful and sophisticated link added to the immortality spell by Esther as a method of further protecting her children.
Rebekah and Kol are the two living Originals with and without a sireline, respectively. As such, all remaining vampires are either from Klaus' unlinked sireline or from Rebekah's sireline, as it remains unknown if Kol has turned any other vampires following his resurrection. There is yet to be any vampires sired from an Enhanced or an Upgraded Original vampire.
Mikael was the first Original vampire destroyed. However, it is never mentioned if he sired any vampires, nor is there any mention of the destruction of vampires in a sudden and mysterious way. It is assumed, given he despised vampires even after becoming a vampire hunter and his preference to feed off of vampires, he most likely never sired any vampires.
It is shown that an Original's spirit has to die in order to kill off their bloodline; even though Klaus' body was supposedly destroyed, his soul was in Tyler Lockwood's body, so his bloodline was unharmed.
However it is possible, that since Klaus' body was not destroyed, that is the reason his bloodline remained intact. This seems to be the case as Esther wished to destroy Rebekah's body after she was jumped into Eva's body.
It is possible to unlink a vampire from their bloodline using magic, as Davina attempted in Season Two of The Originals, but her efforts were sabotaged by Kol.
In Heart Shaped Box, Kol stated that Esther created the sirelines as a further precaution to protect the thing she loved the most, her family.
Kol also revealed that the only way to break the sirelines is with the heart of an unsired vampire. The only one in existence is Hayley Marshall-Kenner. However, through the Unification Ritual, the ability was transmitted to Jackson Kenner.
It is unknown if future vampires would be sired to him, as the spell unlinked existing vampires, nor has he sired any new vampires since it was broken.
According to Lucien, vampires from all over the world heard that an attack was made on Klaus' life by Dahlia. Lucien also stated that each of the sirelines is at war and an attack may be made on either of their lives.
Kol Is the only original who's sireline members and first sired vampire are still unknown.