A musical note is a vibration in the air arriving at a steady rate. Pluck a string and it oscillates hundreds of times a second; the faster it goes, the higher the note sounds. That rate is the frequency, measured in hertz, and the note orchestras tune to sits at Hz.
The octave
Double a frequency and something remarkable happens. Play Hz and Hz together and they do not sound like two different notes so much as the same note in two places. Every musical culture that has been studied treats them as equivalent, and both are called A.
The reason is in the physics. A vibrating string does not produce one frequency but a whole stack of them: the fundamental, then twice it, three times it, four times it, and so on. These harmonics are present in every natural tone and give an instrument its character. When two notes are an octave apart, the upper note's harmonics are already a subset of the lower note's, so the two blend almost seamlessly.
This is why note names run out and start again. The alphabet goes A to G and then returns to A, because after seven letters you have arrived back at the same note, doubled.
Dividing the octave
Between one A and the next, Western music inserts eleven more notes, making twelve equal steps in all. Each step is a semitone, the smallest distance in standard notation, and two semitones make a tone.
On a piano keyboard these twelve are the white and black keys taken together, repeating in a pattern of alternating groups of two and three black keys. That visual pattern is a map: it tells you where you are without counting from the end.
The awkward compromise
The white keys are not evenly spaced. From E to F and from B to C is a single semitone, while every other pair of adjacent white keys is two. The black keys fill the wider gaps, and each takes two names: the note above C is both C sharp and D flat, one step up from C or one step down from D.
That twelve-note division is a deliberate compromise. Tuning by pure harmonic ratios gives gorgeous results in one key and unusable ones in another, so equal temperament spreads the error evenly, making every semitone the same size, precisely times the frequency below it. Nothing is perfectly in tune, everything is close enough, and you can play in any key on the same instrument. With the twelve steps in place, what matters next is the distance between them.