The way I learned it, there are two kinds of pitch perception, relative and absolute. Relative pitch means that, given one note, you can discriminate other notes in relationship to it. Absolute pitch means that you can recognize the pitch of any note all by itself.

I'm a fiddler and I once tested myself for pitch with an audio oscillator. If I listened to a 440 A and then tried to reproduce the sound, I did it quite well, to somewhere within the error of measurement of the equipment. When I tried to produce a 440 A with no audible clues, I was not as good, but not bad, coming within 1 or 2 Hz of 440. These two tests would correspond to relative and absolute pitch, as I learned the terms, respectively. I learned to do both and I believe that I could improve my skill with practice.

I know someone who knows someone with a remarkable sense of pitch. You can place each of your 10 fingers on a piano key and play all the notes simultaneously, and she can tell you what each of the 10 notes is. Sometimes she's not sure which octave a note is in, but she's always sure of the note. She says that each note is associated witha particular color in her mind.