Joe,
The Circle of Fifths really should be called the Cycle of Sevenths or the Cycle of Fourths. It's purpose is to see the relationship in chords when you modulate. If you look at the Circle of Fifths backwards it's more helpful.

IE: C,F,Bb,Eb,Ab,Db,Gb(F#)B,E,A,D,G,and back to C again.

Here's how it's used. C7 leads to F. F7 leads to Bb. Bb7 leads to Eb.....and so on around the cycle.

Many early jazz tunes use a half-cycle or what is known as a back-cycle.

For instance, if you are in the key of C, the song may get to a B7 chord, go to an E7, ....A7.....D7......G7 and then back to C.

A part cycle tune would be Five Foot Two (Eyes of Blue)

C//// E7 //// A7//// //// D7//// G7//// C////............


Some songs use partial cycle..some more. THe modern jazz Toots Thielmann song Bluesette based on be-bop blues changes by Charlie Parker uses the entire cycle. Starting with Bb.

Bbmaj7/// /// Am7b5 /// D7b9/// Gm7/// C7/// Fm7/// Bb7///
Ebmaj7/// /// Ebm7/// Ab7/// Dbmaj7/// /// Dbm7/// Gb7(F#7)///
Bmaj7 /// /// Cm7/// F7/// Bbmaj7 (turnaround) G7+/// Cm7///
F7///

The song doesn't use the chords in exactly the same order as the cycle but pretty consistent.

The Cycle is used in jazz as a basis for understanding harmonies that modulate within a tune.

Frank