The Circle of 4ths (or 5ths)

The Circle of 4ths contains a huge amount of information when we understand it in some depth. 
A songwriter may know nothing about the circle of 4ths but use it all the time in their songs. It is particularly useful in performance, composition, playing by ear, improvisation, arranging teaching and general understanding of music.

After playing music for many years, the circle of 4th's has become second nature as so much music uses the movement laid out in it. For example; C chord often goes to F. G often goes to C. D often goes to G.

These chords can be major, minor, 7th or any other chord type.
'Autumn Leaves' is a popular song.
The chord sequence is:

Take the chord endings off to make the root movement clearer.

Check these out by starting on D on the circle of 4ths.
Follow the arrows clockwise and you'll see that the next note is G then C then F.
At this point the cycle breaks and goes to a B.
Keep following the arrows and you'll see that E is next and and then A.  

Not all songs follow the cycle and rarely do songs follow the cycle through all the notes. 
'Down in Brazil' by Michael Franks is an exception. It begins on A.
Then Am to D7 to G. Then Gm to C7 to F. Then Fm Bb7 to Eb. Then Ebm Ab7 to Db. Then Dbm Gb7 to B. Then Bm7 E7 and then returns to A to begin the cycle again.

We have barely scratched the surface of the circle of 4ths.