Peter's archetypal ability is his unending youth. In Peter and Wendy, it is explained that Peter must forget his own adventures and what he learns about the world in order to stay childlike. The unauthorised prequels by Barry and Pearson attribute Peter's everlasting youth to his exposure to starstuff, a magical substance which has fallen to earth.
Peter's ability to fly is explained, but inconsistently. In The Little White Bird he is able to fly because he -- like all babies -- is part bird. In the play and novel, he teaches the Darling children to fly using a combination of "lovely wonderful thoughts" (which became "happy thoughts" in Disney's film) and fairy dust; it is unclear whether he is serious about "wonderful thoughts" being required (it was stated in the novel that this was merely a silly diversion from the fairy dust being the true source), or whether he requires the fairy dust himself. However, in Barrie's Dedication to the play Peter Pan, The boy who wouldn't grow up, the author attributes the idea of fairy dust being necessary for flight to more practical considerations:
...after the first production I had to add something to the play at the request of parents (who thus showed that they thought me the responsible person) about no one being able to fly until the fairy dust had been blown on him; so many children having gone home and tried it from their beds and needed surgical attention. - J.M. Barrie
In Hook, the adult Peter is unable to fly until he remembers his "happy thought". The ability to fly is also attributed to starstuff -- apparently the same thing as fairy dust -- in the Starcatchers prequels.
Peter has an effect on the whole of Neverland and its inhabitants when he is there. Barrie states that although Neverland appears different to every child, the island "wakes up" when Peter returns from his trip to London. In the chapter "The Mermaids' Lagoon" in the book Peter and Wendy, Barrie writes that there is almost nothing that Peter cannot do. He is a skilled swordsman, rivalling even Captain Hook, whose hand he cut off in a duel. He has remarkably keen vision and hearing. He is skilled in mimicry, copying the voice of Hook, and the tick-tock of the Crocodile. In the 2003 film, the mermaids speak by making dolphin-like noises, which Peter can both understand and speak.
In both Peter Pan and Wendy and Peter Pan in Scarlet, there are various mentions of Peter's ability to imagine things into existence, such as food, though this ability plays a more central role in Peter Pan in Scarlet. He also creates imaginary windows and doors as a kind of physical metaphor for ignoring or shunning his companions. He is said to be able to feel danger when it is near.
In Peter and Wendy, Barrie states that the Peter Pan legend Mrs Darling heard as a child was that when children died, he accompanied them part of the way to their destination so they would not be frightened; he thus resembles the Greek god Hermes in his role as a psychopomp.
In the original play, Peter states that no one must ever touch him (though he does not know why), and the stage instructions specify that no one does so throughout the play. Wendy approaches Peter to give him a "thimble" (kiss), but is prevented by Tinker Bell.