Meet the Volturi

Since we are about to watch New Moon in a few weeks time. Let's get to know the Volturi and their interesting family.

Leaders
Aro, Caius and Marcus are the leaders of the Volturi. They command the Volturi's guard and act as representatives when dealing with other covens. Aro has the ability to read every thought a human or vampire ever had simply by making physical contact with them. Marcus has the ability to sense relationships, as demonstrated in New Moon. Unlike Aro and Marcus, Caius is not gifted.

Aro
Caius
Marcus

The Wives

Mates of the Volturi leaders. The wives are also powerful figures of authority, but only go to trials with their husbands on behalf of extremely important matters, not leaving Volterra but once every few centuries. Didyme had the power to make an aura of happiness, similar to Jasper's gift. Unlike Didyme, Sulpicia and Athenodora are not gifted.

Athenodora (Caius' mate)
Sulpicia (Aro's mate)
Didyme (Marcus's deceased mate)

The Guard

Jane- a high ranking member of the Volturi
Demetri and Felix important members of the Guard
The Volturi Guard is a group of vampires with powerful gifts. The Volturi has been shown to invite only vampires with incredibly powerful gifts that the leaders of the coven see as useful to ensuring that their authority amongst the other covens goes unchallenged. When a vampire is invited into the Volturi, he or she is forced to feel content about joining due to Chelsea's gift to manipulate the emotional bonds of others, usually guaranteeing a swift initiation into the group. Felix and Santiago are the only two ungifted Volturi guards.

The members are as follows:

Afton-unknown talent
Alec-sensory paralysis
Chelsea-relationship manipulation (cannot manipulate emotional ties as easily, eg. Bella and Edward)
Corin-unknown talent
Demetri-tracking sense (known as the greatest tracker in the world)
Felix - (assuming) immense strength
Heidi-unknown talent
Jane-illusion of pain
Renata-protective shield (can form a mental force field to avoid physical attacks, her power is quite similar to Bella's ability to create mental shields to avoid mental attacks.)
Santiago

Former Members

Eleazar: A former member of the Volturi. He was invited into the coven because Aro thought his ability to see the gifts and potential gifts of others was useful for when the Volturi scouted for new members. He resigned after meeting his spouse, Carmen, and joined the Denali Coven.

Didyme: Marcus' wife and Aro's sister. She was murdered by Aro when she and Marcus began to discuss leaving the Volturi.

Other Gianna (Human secretary)

Gianna is only mentioned in New Moon. She had hopes to become a vampire also but Stephenie Meyer has said that Gianna did not survive and was instead killed to sustain the Volturi. There are supposedly more unnamed Volturi and also numerous witnesses, but they are not named in the books.

Desired Members

Carlisle Cullen: Stayed with the coven as a guest. Though he didn't have a special talent that the Volturi saw as useful, they still enjoyed his company. He left due to his reverence of human life, and created his own coven in the United States.

Edward Cullen: Invited several times by the Volturi because of his ability to read minds at a distance. He declined.

Bella Cullen: Aro was impressed with Bella's ability to block mental powers when she was still human. He invited her to join the Volturi when she was still human, and again when she had been turned into a vampire. She declined both times.

Alice Cullen: Invited by the Volturi due to her precognitive powers. She declined. Edward said that Aro never wanted anything more than Alice's powers.

Benjamin: A member of the Egyptian Coven with the ability to physically control the elements. He was offered a place with the Volturi at the confrontation in Breaking Dawn. He declined.

Zafrina: A member of the Amazon Coven with the ability to project illusions into the minds of others. She was offered a place with the Volturi at the confrontation in Breaking Dawn. She declined.

Kate: A member of the Denali Coven with the ability to shock others with an electric current on her skin. She was offered a spot with the Volturi at the confrontation that took place in Breaking Dawn. She, like the rest who were offered, declined.

Renesmee Cullen: Since Renesmee is one of the few half-human vampires in the world, the Cullens assumed that Aro would want her in his "collection." However, while Aro was intrigued by her, an offer for her to join the Volturi was never made.

Shape-Shifters of La Push: During Renesmee's "trial", Aro saw memories of the Cullens' interactions with the La Push wolves. Thinking the Cullens had somehow tamed the wolves to be their pets, he was intrigued by the idea of the wolves working for him as guard dogs. The wolves snarled in response, effectively putting an end to that line of thinking.

Source: Twilight Saga Wikia

Stephenie Meyer's Official Website