Concert pitch is simply a measurement which was set to allow more than one instrument to play together, and still be in tune with each other. For example, a clarinet playing a C note might have a pitch of 525 Hertz, and a piano playing the same C might produce a pitch of 520 Hertz. On their own, these two instruments would sound absolutely fine, but if they played at the same time, they would sound distinctly out of tune!
Concert pitch is usually measured from 440 Hertz, which is an A note. If you hear an orchestra tuning up before a concert, you will hear them tune to this note to make sure they are all at the same pitch.
A piano does not have to be at concert pitch to sound in tune, as long as all the notes are in tune with each other. In fact, a lot of older instruments may be incapable of reaching concert pitch due to the strings being too old. As long as the piano is not being used to accompany another instrument, and the pianist isn’t playing along to a recording, this is perfectly fine.
If a piano is a long way from concert pitch, it is sometimes necessary to tune it twice to get the piano to stay in tune.