By coming across the "essence" of an individual's mind and focusing on it, Demetri can find said individual no matter where they are. He is known as the greatest tracker in the world, far superior even to James. He doesn't need to have physically met the person to be able to track them down; by coming across someone else who had met the target, he can pick up the essence through him or her.
Limitations
Seemingly limited to finding living beings only, Demetri can only link to his targets once he has physically met them or has picked up his or her trail from someone who has met them in the past. It has been confirmed by The Twilight Saga: The Official Illustrated Guide that only strong mental shields such as Bella Swan's can render his power useless.
History
Origin
Though there was no official description as to how his power manifested, it was implied that Demetri was already skilled at tracking as a human, which drew Amun's attention and prompted him to turn Demetri into a vampire.
Post-transformation
The two worked together to develop his ability and in time, Demetri's power was matched by no tracker. This eventually drew the Volturi's notice, and Aro replaced his previous tracker with him, who had a less potent power of tracking.
Ever since then, he had been part of all of the Volturi's important missions. Whenever a coven or individual had committed a crime that drew the Volturi's notice, Demetri was assigned to find them, since no one could ever escape his power.
New Moon
"He's a tracker―a tracker a thousand times more gifted than James was. His ability is loosely related to what I do, or what Aro does. He catches the... flavor? I don't know how to describe it... the tenor... of someone's mind, and then he follows that. It works over immense distances."
Upon leaving Volterra by the end of New Moon, Edward tells Bella and his family that he did not want to touch Aro's hand again, because he did not want him to find out he had discovered that Demetri's tracking ability is mental. Working from this statement from the book, it is fair to assume that thoughts linger or leave behind some kind of trace Demetri is able to pick up and follow. The interesting part of the workings of Demetri's ability is that it would be ineffective on Bella, due to "her brain being wired wrong".
Eclipse
Demetri and the others coming to the Cullens.
Demetri is assumed to have accompanied Jane, Felix, and some of the other Volturi guards under Aro's orders to study and eradicate the Seattle newborn army.
In the movie, Demetri accompanies Jane, Alec, and Felix to Seattle and then Forks, hinting that he tracked the newborn army to their locations.
Breaking Dawn
In Breaking Dawn it is also commented that if Demetri's power did not make him the greatest tracker in the world, Aro would have replaced him by now, as Aro doesn't surround himself with "second best".
The Cullens believed that, because of his skills and capabilities, killing Demetri would be one of their main priorities if it came to open war with the Volturi. Renesmee, Bella and Edward's daughter, was the primary target on the Cullens' side, and they knew that if somehow Renesmee did escape with her life, Demetri would be sent to find her and bring her in for "justice".
"Alistair can track both people and things. He can sense the general direction of whatever he is looking for, but it takes him a long time to narrow this feeling down to a specific location. If whatever he's seeking is on the move, he may never catch up."
Alistair is the only vampire known to be able to track inanimate objects in addition to living beings.
His power works by focusing on a target to find its general direction; he can also narrow down a location and the distance between them. It is described as a kind of a magnetic pull toward the target, though it is possible to avoid it.
Given that this power can track objects in a magnetic form, it is likely to be unaffected by Bella Swan's shield.
Limitations
Though useful, this ability is more limited than those of James's and Demetri's. Alistair mostly feels the general direction of the object he focuses on, and it takes some time to narrow his feeling down to a specific location. If whatever he is seeking is on the move, he may never catch up.
History
Origin
Alistair had a passion with animals and was a fairly skilled hunter.
Post-transformation
By becoming a vampire, this trait manifested into a sort of power that allowed him to track any object he focused on.
Breaking Dawn
"Alistair was a tracker, though not nearly as precise and efficient as Demetri. Alistair just felt an elusive pull toward whatever he was seeking."
Edward describes Alistair's ability to Bella when he first arrived in Forks, though begrudgingly, to witness for the Cullen family. Alistair flees Forks after sensing the numbers of the Volturi and their witnesses coming to confront the Cullens.
Unnamed tracker
According to The Twilight Saga: The Official Illustrated Guide, the Volturi had another, less powerful tracker before Demetri joined them. There is no confirmation as to how this tracker's power worked nor what happened to him after he was replaced by Demetri.
Alice Cullen has the power of foresight. Though this power is not the same as tracking, it can be used in that method to anticipate a target's moves and the direction it is headed.
Edward Cullen has the power of telepathy. Though this is different and far less efficient in tracking, it can be used to keep tabs of a target's thoughts and use that as a means to find their location.
