Pitch is a smooth slope: between any two notes there are endless in-between frequencies. Music chops that slope into steps, the way a staircase turns a hill into something you can climb. Different traditions cut the steps differently, and each cut gives music its own flavor.
Twelve notes and their names
The Western system divides the octave into twelve different notes. Seven of them have letter names, A, B, C, D, E, F and G, which repeat in every octave. On a piano these are the white keys. The five black keys fill the gaps between them. A half step, also called a semitone, is the smallest distance in the Western system: the step from one piano key to the very next key, black or white. A whole step is two half steps. From C to D is a whole step, because a black key lies between them. From E to F there is no black key, so that step is only a half step.
The black keys are named by their neighbors. A sharp raises a note by a half step, and a flat lowers it by a half step. The black key between C and D can be called C sharp or D flat. Two names for the same pitch, such as C sharp and D flat, are called enharmonic.
Scales
A scale is a set of notes arranged in order of pitch that a piece of music draws on. Most Western melodies use seven of the twelve available notes, chosen by a pattern of whole and half steps. The major scale follows the pattern whole, whole, half, whole, whole, whole, half. Start on C and climb the white keys to the next C, and you will produce exactly that pattern: do, re, mi, fa, sol, la, ti, do.
The natural minor scale uses a different arrangement, whole, half, whole, whole, half, whole, whole. Start on A and play only white keys up to the next A, and you hear it. A third common type is the pentatonic scale, which has five notes per octave. Play only the black keys of a piano and you have one. The hymn tune of "Amazing Grace" uses essentially one.
Keys and home notes
Most music feels as if it lives somewhere. The tonic is the home note of a scale, where a melody feels at rest. A key is the combination of a tonic and the scale built on it: "C major" means C is home and the notes of the C major scale are the main material. A key signature at the start of written music lists which sharps or flats the key needs, so the player does not have to be told each time. Changing the key of a whole piece, to suit a singer's range for example, is called transposing.
Tuning the twelve steps
If you tuned every interval to the pure number ratios of the overtone series, you would find that the steps do not quite add up: a piano tuned perfectly for C major would sound sour in keys far from C. The one that became standard on pianos by the late 1800s is equal temperament, which makes every half step exactly the same size, so music sounds equally usable in every key. The cost is small: most intervals in equal temperament are very slightly out from their pure versions, by amounts many listeners never notice.
Other ways to cut the octave
The twelve-note system is one choice among many. Arabic and Turkish music organizes melody into maqams, some of which use intervals that fall between the Western half steps. Javanese gamelan music uses scales whose steps fit no piano. In Indian classical music a raga is more than a scale: it is a framework with rules for ascending and descending, characteristic phrases and associated moods. None of these is a "wrong" version of Western tuning. Each is a complete and sophisticated system, and learning one helps you hear the others more clearly.
Worked example: moving "Twinkle, Twinkle" to a new key
Play "Twinkle, Twinkle, Little Star" starting on C. Using letter names, the first phrase is C, C, G, G, A, A, G, then F, F, E, E, D, D, C. Every note is a white key, because the tune lives in C major and C major has no sharps or flats. The melody begins and ends on C, the tonic, which is why the last note sounds like arriving home.
Now suppose a singer finds that too low, and you want to start on D instead. Shift every note up by the same distance, a whole step: C becomes D, D becomes E, F becomes G, G becomes A and A becomes B. But E must become F sharp, because the D major scale, following the whole, whole, half pattern from D, needs F sharp rather than F. Try playing a plain F on "how I wonder" and you will hear at once that it sounds wrong. D major's key signature has two sharps, F sharp and C sharp, but this tune never uses the seventh note of its scale, so only F sharp appears. The experience shows why key signatures exist: they tell you in advance which black keys a key will need.