The vast majority of people who exhibit absolute pitch appear to have this skill as an innate ability. It is not learned, but they are apparently born with the ability. A lot of research has been done regarding whether there is a genetic basis for absolute pitch, but this has been inconclusive. Research provided in the Journal of the Acoustical Society of America (JASA) has shown that there is a higher incidence of people with absolute pitch who were born or brought up in East Asia (see the link in Resources), but it's unclear why.
Training courses claiming to teach the acquisition of absolute pitch have been around for more than 100 years. But there has never been a scientifically documented case of anyone who was successfully taught this skill as an adult. There is better evidence (shown again in the Journal of the Acoustical Society of America) that children taught within a specific window of opportunity, usually from 2 to 4 years old but sometimes extending up to 6, may be taught to acquire perfect pitch.
Many brands of pitch training software are available. There are also many courses offered on CD. Often, these consist of games where so-called "target pitches" are played in different musical contexts, challenging the listener to pick out the common pitch. Some methods teach using atonal contexts to minimize the use of relative pitch and sharpen listening skills. Some courses emphasize the different "chroma" or tonal colors associated with each degree of the scale.
While many people who possess absolute pitch are musicians, most professional musicians do not have the ability. It is not necessary for a high level musical career. In fact, some musicians cite disadvantages to possessing perfect pitch. For example, if a piece has to be played or sung in a non-standard pitch or transposed into a different key, it may sound "wrong" to their ear.
While there is no evidence for the success of absolute pitch training courses or programs at least for adults, any course that is well designed will increase musical listening skills, improve recognition of harmony and harmonic progressions and certainly improve the trainee's relative pitch skills. Musicians looking to develop this skill would be well advised to temper their expectations.