The circle of fifths puts the twelve keys in order so that neighbours share the most notes. Once you can read it, you can find the chords of any key, see where a song's odd chord came from, and move a progression to any key without thinking about individual notes.
Clockwise is up a fifth. C → G → D → A. Each step adds one sharp (or takes away one flat).
A key's chords sit together. IV, I and V side by side on the outside, ii, vi and iii right under them. Above, C major's six chords are highlighted: F C G outside, Dm Am Em inside. The seventh chord, vii° (B°), is the leftover.
The inner ring is the relative minor. Each minor has exactly the same notes and chords as the major above it; only the home chord changes. A minor and C major share everything.
Neighbours are smooth key changes. Keys next to each other share six of their seven notes. Jumping across the circle is a bold change.
Counterclockwise is the strong direction. Roots falling a fifth (one step counterclockwise) is the backbone of most progressions: V to I, ii to V, vi to ii. The pull is strongest when the first chord is a dominant 7th. A dominant 7th on that clockwise neighbour is a secondary dominant: E7 → Am, A7 → Dm, D7 → G in C major.
Sus, add9, 6 and 7th chords don't get their own spot. They are colours of the chord in the same slot: Dsus4 lives where D does.
Chords borrowed from outside the key
Rock and pop borrow chords from the parallel minor: in C major, B♭ (♭VII), Fm (iv), A♭ (♭VI) and E♭ (♭III). They sit just counterclockwise of the key's cluster. Hey Jude's coda (F E♭ B♭ F) and Sweet Child O' Mine (D C G D) are the ♭VII–IV–I move.
A map of the 12 keys. Each step clockwise is up a fifth (C, G, D, A...) and adds a sharp or drops a flat; each step counterclockwise is down a fifth and adds a flat or drops a sharp. The inner ring shows each major key's relative minor.
How do you find the chords in a key with the circle of fifths?
Find the key on the outer ring. The keys on either side of it are its IV and V chords, and the three minors directly below those three are its ii, vi and iii. That gives six of the seven chords; the seventh is the diminished vii°.
Why is it called the circle of fifths?
Because neighbouring keys are a perfect fifth apart, and after twelve fifths you arrive back where you started.