It's believed to be inborn or somehow acquired spontaneously at a very early age. There are products that claim to teach it, but there's no scientific evidence. Years ago I tried a few, and my impression was that it's mostly snake oil, and even if some rudimentary progress is possible, it's definitely not worth the effort, better expended on improving relative pitch and other aspects of musicianship. Wikipedia agrees:

From http://en.wikipedia.org/wiki/Absolute_pitch "[...] there are no reported cases of an adult obtaining absolute pitch ability through musical training; adults who possess relative pitch, but who do not already have absolute pitch, can learn "pseudo-absolute pitch", and become able to identify notes in a way that superficially resembles absolute pitch. Moreover, training pseudo-absolute pitch requires considerable motivation, time, and effort, and learning is not retained without constant practice and reinforcement."

It is likely some amount of both.

Studies have shown that there is a higher prevalence of people with perfect pitch in countries where tonal languages--languages in which the same series of sounds made with distinct voice pitches can denote entirely different words.

However, the percentage of people with perfect pitch remains tiny, which is not what I would expect if it was due to training alone. It seems like it might be helpful enough for musicians that we'd have turned it into a method by now if training alone reliably produced good results.

I'm fairly certain that it is one of those things you either can hear, or cannot hear - it requires a certain sensitivity in the ear, and not all people have that. Not all people that can hear the difference have trained it to a degree where they're conscious about it, so for some people it's possible.

Around the age of 15, I acquired something an awful lot like perfect pitch: I can identify and (conversely) produce seven particular notes. n=1.