Doppelganger

A doppelganger is a person who looks exactly the same as a specific person who lived (or was born) in a previous generation. While the frequency of their appearances may vary, they are arguably the rarest supernatural species.

The reason of a doppelganger's existence depends on the mythology of the series they appear in.

Post Season 3 Series
In this series, the only existing doppelgangers are descendants of Tatia. While there could be others that aren't introduced, canonically she has only two doppelgangers over the span of a thousand years: Katerina Petrova and Elena Gilbert.

The reason doppelgangers exist is because Esther [somehow] sent Tatia's spirit to the Dark Dimension after using her in a spell that turned her children into immortal vampires. As she was the source of magic that turned the original family into vampires, her survival was crucial. Since as a spirit, she would live safely forever in Dark Dimension, allowing Esther's children the same courtesy of immortality.

However, her spirit being in another dimension caused an imbalance. The nature had Tatia's body, but not her soul. To fill in that blank, every once in a while nature created copies of Tatia. But every new spirit came with a body, so the balance could never be restored. Therefore the doppelgangers were going to keep appearing every few generations as long as the bloodline continued.

The cycle eventually ended in the series finale, The Final Conversation. After learning the last original vampire died, Tatia decided to return to her home planet, realizing there was no point in staying in Dark Dimension anymore. With the missing spirit returning to Earth, the balance was restored, meaning nature will not create another doppelganger, effectively ending the species forever.

Stefan's Diaries
In this series, doppelgangers are of two different people: Silas and her unnamed one true love. It is revealed that Stefan was Silas' doppelganger, and Tatia, Katerina Petrova and Elena Gilbert are his lover's. Around two thousand years before today, Silas tricked a powerful witch Qetsiyah to help him create a spell that would grant him and his lover true immortality, meaning there would be no possible way to kill them and they would live together forever.

Qetsiyah discovered the deception. To punish him, she created a cure for his lover that turned her back to human, and then killed her, condemning Silas into a lonely and eternal life.

His true immortality was not something nature would approve. As it intended, every living creature had to have a way to die. But since Qetsiyah's spell made sure that there would be no way at all for Silas to die, nature was forced to create mirror images, shadow selves, -otherwise known as doppelgangers- of him that can die.

Controversially, nature continued to create doppelgangers of his lover despite she was cured and killed, making the reason of their appearances unknown.

The doppelgangers, along with every other supernatural species, became extinct at the end of series. After Ewa cast a spell that would kill every supernatural species in the world, for survival, witch covens from all around the world joined forces and destroyed the magic on the planet. This turned everything supernatural into humans, effectively causing the absolute extinction of every supernatural species, including doppelgangers.

Powers & Abilities
Doppelgangers do not possess any special abilities that makes them any different than a normal human. The only difference is, since they are very rarely occurring in the nature, their blood provide more magical power than any other species, making them the only way to achieve certain magical accomplishments.

So far, doppelganger blood was used and/or intended to be used for:
 * Creating vampires: It was revealed that the blood of Tatia, the first doppelganger, was used by Esther to create the first vampires in history. The mythology varies from series to series, but in each case, the magical power that can only be provided by a doppelganger's blood has been crucial.
 * The Hybrid Curse: When Esther realized Klaus' first kill as a vampire triggered his werewolf gene, therefore making him a hybrid, she created a curse that suppressed his werewolf side, making him no different than his vampire siblings. Since the creating of this peculiar curse required the same magical blood that made his becoming a hybrid possible in the first place, Tatia's blood had to be used in the process.
 * Breaking the Hybrid Curse: Requiring doppelganger blood to break this curse was a repetitive necessity rather than magical. A doppelganger's blood was used to create the curse, therefore only doppelganger blood could break the curse.
 * Transitioning hybrids: This is a situation specific to vampire-werewolf hybrids. Normally when a werewolf is turned into a vampire, their nature denies this change and whether they feed on blood to complete the transition or not, their metabolism eventually shuts down and the hybrid's transition into a vampire fails. However, by chance, Klaus discovered that blood from a doppelganger would provide the means to make this transition successful, making Tyler Lockwood his first successfully sired hybrid.
 * Creating hybrids of various species: While a werewolf-witch hybrid may naturally be born, vampire hybrids of different species are virtually impossible to create. In the case of witches, they lose their magic, and in the case of werewolves, they simply die. However, using doppelganger blood in the creation of these hybrids provide a way around this.
 * "Klaus" : This was never confirmed, but it is arguable that Klaus was only able to survive his transition into a vampire because he was turned by (and fed on to complete his transition) a doppelganger's blood. Another case may be that he was able to become a vampire because his werewolf gene wasn't triggered by then. This was never clarified.
 * "Pure Blood Angels" : In the 4th season of Post Season 3 Series, pure blood angels came to Mystic Falls to use doppelganger blood to sire successful hybrids with other species. In their case, making hybrids with other species caused the pureness of the born hybrid's angel side, and they hoped to make hybrids that were still pure blood angels by using doppelganger blood in an unspecified magical way.
 * "Isla Kennedy" : Isla was originally a witch. She decided to become a vampire, but she also wanted to keep her magic. This was impossible, as it was stated many times in the series. One could only be a vampire or a witch, but never both. Isla, however, by chance, found a way to make it possible. Her plan was to use the cure [for vampirism] in Katherine's blood to trick her spirit into thinking she was still a witch while her body acted as a vampire, making her both a vampire and a witch. Little did she know that, while the cure was also crucial, it was Katherine's doppelganger blood that made the process successful.
 * Providing infinite magical power: In Post Season 3 Series, Inari has been looking for a way to stay on Earth without losing her magic. Like all other kitsunes, Earth was absorbing her magic, making her weaker in time. As a way to prevent that, Inari wanted to use the doppelganger in a sacrificial ritual. This was prevented by the Mystic Falls Gang, so whether it would be possible remains unknown.
 * Unknown: In the 4th season of Stefan's Diaries, a Massachusetts based witch coven will use Stefan's human blood for a villainous purpose.
 * Supernatural extinction: This was rather an indirect use of doppelganger blood. In the 6th season of Stefan's Diaries, the witch Ewa vowed to destroy the supernatural world. To achieve this, she wanted to use the doppelganger Katherine. But when she realized that she was previously a vampire and therefore is useless, she instead kidnapped her son Stefan Delaney. Stefan, despite not being a doppelganger, still belonged to the bloodline, therefore was able to provide the magic that Ewa needed to launch her spell.