Concert pitch means merely that the A note above middle C is vibrating at exactly 440 times per second. Assuming that the piano is in tune with itself, the whole piano is at concert pitch.
Sometimes in older pianos, the frame will not take the strain of concert pitch (the higher the pitch, the more tension on the strings and frame). This means that it has to be tuned lower, flatter, or down, which is where, with the piano in tune with A above middle C, The A is vibrating at fewer than 440 times per second.
A common pitch for older pianos is one semitone down (A=415); this means that if you strike the
C key it will actually make the sound of B, and if you strike an F key it will make the sound of E, etc. With this tuning, you cannot use the piano to accompany other instruments unless you transpose all the piano music back up a semitone (which is a feat done only by rare musicians).
It also plays havoc with people who have perfect pitch, as their hands tell them one thing but their ears tell them another!