Each of the special kids has a psychic ability of some kind, except for Jake who has enhanced strength
It is unclear whether the abilities came from the Demon or if they already had them
Sam's abilities seem to have stopped now
Some demons appear to have psychic abilities, such as mind-control and telekinesis
The ability to electrify through touch. Scott Carey had this ability and had used it (at the Yellow-Eyed Demons insistence) to kill a cat.
Andy Gallagher and his twin brother Ansem Weems both have the ability to control the minds of other people. At the start, Andy is only able to control minds through voice commands. It is interesting to note that both twins have the same ability.
Both Sam and Ava have premonitions of future events, although Sam's are accompanied by extreme headaches. Most of the premonitions that Sam has are connected in some way with the Yellow-Eyed Demon.
With practise, Andy Gallagher taught himself to project images into other people's minds. He used this ability to send an image of their location to Dean when he and the other special kids were trapped by the Yellow-Eyed Demon.
Max Miller was able to move things just by thinking about them. On one occasion, when under extreme pressure, Sam was able to do the same, but hasn't been seen to do so since. The Yellow-Eyed Demon used telekinesis on many occasions.