In the Twilight universe, some vampires gain no powers when they turn (you know, except for diamond-hard skin, strength enough to lift a car, regeneration, and being able to run at 100 miles per hour).
On the other hand, some gain additional powers. Edward can read minds. Alice can see the future. Chelsea can manipulate people's emotional bonds.
This is because the nature of vampification is to enhance the qualities that someone already has. Those humans with latent powers get full-fledged powers when they turn, whereas humans without such powers simply gain enhanced personality traits.
How common are vampires with special powers? Clearly most of Carlisle's coven has such abilities, as do many of the Volturi, but is this the norm, or are powers more rare among vampirekind as a whole?