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.
Mikael was the first Original destroyed, but it is not mentioned who was turned by him, nor is there any mention of the destruction of vampires in mysterious ways.
As Mikael despised the rest of the vampire species, even becoming a vampire hunter known as the Vampire Who Hunts Vampires, and his preference to feed off vampires, Mikael most likely never sired vampires from his blood.
Kol was the fourth Original to be impaled with a White Oak Stake and the third killed by it.
It is known that his bloodline consisted of at least thousands of vampires as the death of his bloodline managed to complete Jeremy's Hunter's Mark.
In the books, the origin of vampires is not established, but may have been born of nature like the other supernatural beings, as the bloodline of Klaus was not destroyed along with him.
The Original Family acts as a pillar of the vampire species. If destroyed, the species itself will die out.
Klaus did actually turn Mary Porter. This was revealed by Rebekah and Elijah after Klaus' "death." Previously, Klaus had only claimed he started the Salvatores', Tyler's, and Caroline's bloodline, though Damon suspected he had only done so to ensure they would not kill him (though Damon did not mind doing so if it just meant killing Tyler).
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.
In the third season of The Vampire Diaries members of the first generation of humans in two branches of the whole tree turned by Originals were shown: Mary Porter who was turned by Klaus and Sage who was turned by Finn.
In the third season of The Originals, more members of the first generation of vampires in their respective branches were shown: Lucien Castle and Tristan and Aurora de Martel by Klaus, Elijah and Rebekah, respectively.
Before Tyler was resurrected as a werewolf, Klaus remained the only hybrid of his bloodline until Hope, who is of Klaus' blood, sired Hayley.
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.
Klaus, Kol, Elijah and Finn are the only living originals without a sireline.