Aural training is training to give the ability to recognize and understand the music one hears. This can take the form of transcribing music into notation by ear or by hearing musical functions. Aural Training could also be used to identify formal structures on both large and small scales. One's ability in aural training varies greatly from person to person. For some it comes rather naturally and for others it is much more difficult, regardless of whether or not they are a musician. For those with perfect (absolute) pitch, some aspects of aural training are certainly made easier, but not all.
By focusing one aspect of the music at a time (i.e. rhythm, melody, bass line, etc.) one can, with proper training, transcibe the music they hear, into musical notation. This is a very common practice amongst jazz musicians as they transcribe solos recorded by other jazz musicians so as to develop their own improvisational abilities.
In tonal music, different chords have different functions. It is possible to hear these different functions so as to have a better idea of what is happening in the music. Beyond function, even hearing the quality of the chord can be important and be used by musicians. These more general aspects tend to be used by more musicians (classical, jazz, etc.) than transcription.
On a large scale one can train themseleves to hear the different formal sections of a piece in any music. This could be through hearing the difference between verse and chorus in a popular song or hearing the the difference between the Exposition, Development and subsequent Recapitulation in classical sonata form. On a smaller scale, one could hear the differences between the Main Theme, Transition, and Subordinate Theme or even the structures in a single phrase (i.e. basic idea, contrasting idea, continuation, etc.)
The ability to remember the particular quality of pitch classes well enough to so as to recognise them on hearing. It is not a rare ability among musicians.