Even though most of the time vampire-werewolf hybrids are referred to as "hybrid", hybrids of other species were seen too throughout the fanfictions. Most notable hybrids are listed on this page.
The method to create these hybrids are turning werewolves into vampires. Process requires blood from the original hybrid Klaus to turn the werewolf, and blood from the doppelganger to complete the transition successfully.
The only known vampire-werewolf hybrid that didn't turn with Klaus' blood was Klaus himself, who still needed blood from the original doppelganger, Tatia, to survive his transition.
Other vampire-werewolf hybrid in the series was Tyler Lockwood.
This type of hybrid can only be born, as witchcraft and the werewolf curse can only be taken through bloodline. Typically, one parent has to be a werewolf, while the other has to be a witch.
Although it is possible that this type of hybrids are plenty, only known one is Klaus, before he was turned into a vampire and lost his witch abilities.
This type of hybrid was introduced in the second season of Stefan's Diaries.
Typically, when a witch becomes a vampire, they lose their witch side. It was explicitly stated that one can only be a witch or a vampire, but never both. However, a powerful witch called Isla Kennedy managed to brake this rule by using dark magic and blood from the doppelganger, Katherine Pierce.
After a process which is not cleared yet, Isla turned herself into a vampire, but managed to keep her witch powers, thus became the first and only vampire-witch hybrid in history.
This type of hybrid possesses all witch and vampire abilities, including immortality.
Also known as the "Ultimate hybrid", this type of hybrid is virtually impossible to create. However, Klaus, a former witch and vampire-werewolf hybrid, somehow impregnated Hayley, a werewolf, which allowed the birth of a hybrid child, Hayden Mikaelson, who possessed vampire, werewolf and witch abilities.
Even though Hayden's aging only allows her to have limited access to her vampire powers, such as limited speed, strength, healing, mind compulsion, etc., she is still too strong and fast for a normal human.