The circle of fifths arranges the 12 keys so neighbors share all but one note: it is the map of key signatures and smooth key changes.
Take the 12 keys and arrange them in a circle so that each step clockwise adds one sharp: C, G, D, A and so on. That circle is the circle of fifths.
Use it to answer two questions instantly: how many sharps or flats a key has, and which keys are close relatives (the neighbors, plus each key’s minor twin).
Clockwise steps rise a perfect 5th and add a sharp; counter-clockwise steps add flats. Sharps always arrive in the order F#-C#-G#-D#-A#-E#-B#, flats in the mirror order, so a signature is a single glance.
Inside each key sits its relative minor (three semitones below the major tonic) sharing the same signature. Neighboring keys share six of their seven notes, which is why modulations to the dominant or subdominant sound smooth and distant keys sound dramatic.
Practical uses: naming any key signature, choosing closely-related keys for a key change, and seeing why the ubiquitous ii-V-I motion feels like walking home around the circle.
The circle exists because stacking perfect 5ths cycles through all 12 pitch classes before returning home: 12 fifths span exactly 7 octaves in equal temperament. That closure is a property of 12-tone equal temperament; in pure tuning the circle does not quite close (the Pythagorean comma).
Harmonic motion by descending fifths (vi-ii-V-I) is the engine of functional harmony: each root resolves to the next the way a dominant resolves to its tonic, which is why so many progressions read as arcs around the circle.
Reading tip: for sharps, the key is a semitone above the last sharp; for flats, the key is the second-to-last flat. One flat alone is F major, memorized.