Thanks cwilbur for clarifying—I just saw this question, and was somewhat surprised by all the confusing (and flat out wrong!) answers.
here’s an obvious link to give: http://en.wikipedia.org/wiki/Absolute_pitch
To follow up with one point, in my experience, concert pitch would then refer to the B-flat the clarinetist is playing—not the “C” that they’re reading (or thus thinking of it as), but the actual note that’s sounding. So if a conductor says “let’s tune to concert F” everyone knows on their instruments what they need to play to produce the F (so for instance on Horn, which is typically an F instrument, meaning it sounds a fifth lower, that would be a C). Essentially they are NOT playing different notes, but have different names on their instrument for the same actual pitch (due the history as cwilbur described).
And yes, perfect pitch is somehow innate, and relative pitch is learned. But Gail, relative pitch can be worked on to a point of essentially being the same as perfect pitch in practice—- if you do exercises like playing an A above middle C on the piano or a pitch pipe (or whatever you have) every morning, or several times a day for that matter, you’ll start to be able to hear it, in your mind—you can guess at it a moment before you play, and over time your guesses will probably get closer and closer. Chances are that if there are songs you hear (or sing) over and over again (in the same key), you could start singing them on your own roughly in the right key—maybe the same with your alarm clock or microwave beep or another pitch you hear often—if you find out what that key is, you’ll have a reference for it (oh, my kitchen timer dings an Eb!). Anyway, there are many ways to work on this, but as you work on intervals and being able to tell where certain notes fall in your vocal range and so on, you can refine it to be able to do just what you described. Now, if what you’re describing is when a singer finds the A from another pitch (since you wondered if they hear the intervals, I wonder if that’s what you mean – which is not necessarily perfect pitch, as you now know)—then that really is just a matter of working on intervals. In my experience much easier than developing a sense of absolute / perfect pitch! There any many aural skills or musicianship books which give exercises for this, and again, an easy way might be thinking of songs you know—the beginning of Happy Birthday is a whole step (or second), Here Comes the Bride starts with a fourth, Twinkle Twinkle Little Star starts with a fifth, .... even hard ones can be found: Maria starts with a tritone (between a fourth and fifth – typically one of the hardest intervals to pull out of nowhere).
I’m glad you asked this question because for the first time I’ll be teaching a musicianship class, starting in just a few weeks, so I need to dust off my thoughts on this and get together easy descriptions, lesson plan with exercises, etc!! I had very good aural skills training at Oberlin Conservatory, but now years later I’m a bit nervous to suddenly teach an advanced-level undergrad college class!