A person with perfect pitch will be able to, at minimum, know when a piece isn't played in its original key. Most people with perfect pitch will also be able to identify the key a piece is played in (called passive perfect pitch), and some may be able to sing a C-sharp when asked (active perfect pitch). Usually, people with active perfect pitch will not only be able to identify a note, but recognize when that note is too sharp or too flat. Active perfect pitch possessors are about 1 in every 10,000, most of them starting music training before the age of six. However, it is easier for someone with perfect pitch to learn active perfect pitch than it is to learn from scratch.
Persons with perfect pitch will seem annoyed or unnerved when a piece is transposed to a different key, and will have difficulty transposing music without manually calculating intervals between known pitches. They may feel that such a piece does not have the intrinsic beauty of music, and in some cases will be physically uncomfortable; cases are known of musicians who had to tune every instrument or they would actually feel sick. They may have a harder time developing relative pitch than others, and for many musical tasks like transposition, perfect pitch can actually be a hindrance.
Mozart, Leonard Bernstein, and Paul Shaffer are three musicians known to have perfect pitch. However, Stravinsky, Ravel, and Wagner never had perfect pitch. It's worth mentioning here the common mistake of associating perfect pitch with musical genius; many with perfect pitch do not work in music, and it can be a liability when learning to play an instrument (especially for those for whom out-of-tune music is uncomfortable).
Perfect pitch is not limited to the realm of music, or even to humans. Songbirds and wolves have exhibited perfect pitch. In fact, studies indicate that perfect pitch is more a linguistic ability than a musical one. Perfect pitch is an act of cognition, needing memory of the frequency, a label for the frequency (such as B-flat), and exposure to the common range considered a note (a note, in modern tuning, can vary in its exact freqency). It is directly anagolous to recognizing colors. And while most people can recognize and name the color blue by its frequency, only those who have exposure to the names of musical tones, usually musicians, will be likely to identify a middle C. In addition, perfect pitch is more common among speakers of the tonal languages Mandarin Chinese and Vietnamese, which depend heavily on pitch-shifts for meaning. Such cultures have very few "tone deaf" people.
Is perfect pitch an inborn talent or a learned ability? Some scientists believe its preponderance in musical family lineage is due to genetics and are already trying to map the gene for it; others believe most humans don't typically develop this ability because there's no social use for it, and are trying to teach adults how to get it. The debate is certainly not settled yet, as the significant data on this highly specialized ability is quite scarce. It is nevertheless becoming increasingly apparent that people can acquire perfect pitch (at least for single instruments) through learning.
It should be noted that many musicians, and probably most jazz musicians, have quite good relative pitch, a skill which can definitely be learned. With practice, it's possible to listen to a single known pitch once (from a pitch pipe or a tuning fork) and then essentially have perfect pitch for a short period of time, simply by comparing the notes you hear to the known pitch in your head. Unlike true perfect pitch, this skill can also be adjusted up and down as needed.