Historically, the piano tuner tunes by ear. The traditional method is based on a single reference note, now the international A (440hz) emitted by a tuning fork. It consists in comparing this note to its counterpart on the piano. By combining different notes, we listen to the intervals formed between them.
One tool that can be of great use to us today is an electronic tuner. It’s a bit like the one guitarists use, but much more precise. The advantage is to have a stable reference from note to note which allows tunning to be reproduced identically from year to an other.
It’s actually a bit more complicated than that. First, the auditory tuner will listen not specifically to the notes, but to the silence hidden between notes. When the pitch of one note is close enough to another, but not exactly the same, interferences occurs in the sound. What is called a beat is created. The volume is fluctuating more or less quickly. We then “hear” a succession of absences of sound which scrolls through time. Where it gets tricky is that the tuner listens to these interferences with the fundamentals (the notes we usually all hear) and harmonics which are much more discrete sounds that naturally form around the root notes to precise notes.