Piano Tuning has a simple explanation as well as a longer one.
The simple one is that piano tuning is the process of adjusting the tension on the strings so that the piano is “in tune”, meaning at a standard pitch, usually A=440, and is in tune with itself…meaning that the various intervals are all correct and sound pleasant.
The longer story is, well, longer.
Perhaps first, you should read my white paper titled Equal Temperment,
I cannot possibly give a full explanation to the simple question of “what is tuning” here but for the truly interested I can refer you to 2 excellent books.
The smaller one is called On Pitch by Rick Baldassin and is available through the Piano Technicians Guild. This 100 page book covers the technical aspects of tuning a piano.
The larger one is called The Craft Of Piano Tuning by Daniel Levitan and is available on Amazon. This 230 page book covers the same basic content but in much more detail, including sections on the process of tuning, the tuning hammer, tuning pins and the like.
Both books are dense and somewhat difficult to follow, because the process of correctly tuning a piano is difficult.
Piano tuning has a couple of characteristics that separate it from wind or string instrument tuning. The first is that the player has no control of the pitch. With a wind or string instrument the player can subtly adjust the pitch while playing but the pianist is stuck with how ever the tuner left the instrument.
A piano also has a much wider range than any other instrument, meaning that the piano has to be in tune with itself allowing the pianist to play a chord in the low bass and in the high treble and these pitches must all interact pleasantly and correctly. And, the piano can play multiple notes at a time. The clarinetist does not have to worry about how a perfect 5th interval sounds, but the pianist (and tuner) has to make a perfect 5th in the bass fit with a perfect 5th 4 octaves higher….a surprisingly hard task.
You have probably heard the term “overtones” or “harmonics” or “partials”. The problem with pianos is that the stiffness of the steel in the string throws this sequence of overtones off. So, to get a piano to sound in tune throughout it’s huge range, the tuner must compromise and put the piano actually precisely out of tune.
These harmonics or subdivisions of a pitch are the crucial aspect of piano tuning. The piano tuner is actually tuning individual “coincindental” harmonics. To give an example, when I am tuning G4 to G3, I am actually listening to the pitch D6, the 5th partial of the lower G and the 3rd partial of the higher G. This is called, oddly enough, a 5:3 octave. I want these 2 D6’s to be in tune and I’ll manipulate the G4 tuning pin until they are beatless. This is called a 5:3 octave, which is different from a 6:4 octave. Confusing? Good! Piano tuning is hard!.
Setting the temperment, or the initial octave (usually f3 to f4) is the first step. Octaves in piano tuning are always slightly wider than perfectly beatless and the width of this initial octave sets the stretch for the whole piano. Within that octave, the other notes must fit with clean 4th and 5ths and evenly accelerating M3rds.
Maybe you should get one of the books…!