WCT composers love circles of fifths because they give the impression of movement without actually going anywhere, reinforcing the key and creating a satisfying varied repetition.
Each chord in the sequence is a fifth below (or fourth above) the previous one. A complete circle of fifths, such as in the example below, starts at I and then goes all the way round in fifths until it arrives back at I again.
A circle of fifths is usually associated with a sequence as it consists of the first two chords (a falling fifth) repeated down a step each time, as shown by the brackets. This means that once you have written the melody and texture for the first two chords (in this case a bar), you can simply repeat it down a step until you arrive back at I (an even simpler type of harmonic sequence is a chain of first inversions).
Mozart, Piano Sonata in C major, K545, first movement.
In his piano sonata K332, Mozart creates contrasting harmonic colour by moving from C major to C minor and going around a circle of fifths in that key. Notice how he speeds up half way through creating a hemiola from chord VI to add to the momentum. As is often the case in a minor-key circle of fifths, Mozart does not raise the 7th when on chord vii so we get a Bb major chord rather than B diminished.
Mozart, Sonata K 332, 1st movement
In this second example, this time in the key of A minor, you will notice that Vivaldi does not sharpen the seventh on VII and III. This avoids VII being diminished (so we get a G major chord instead of a G# diminished) and III being augmented (so we get C major instead). It is quite common to miss out the sharpened seventh in minor keys to avoid awkward chords in this way.
Vivaldi Concerto Op. 3 No. 8, first movement
A more sophisticated way of using the circle of fifths is as a modulating device:
This first example is very simple. The music moves from C# minor to E major but treating the tonic chord of C# as vi in E major and then continuing in this new key up to a cadence:
Vivaldi Allegro from ‘Spring’ (minor to relative major)
This second example is a bit more complex. The music starts off modulating by making each two chords a perfect cadence in a new key a tone down (first g minor then F major). After two rounds of this, there is a pivot to a longer cadential progression from vi round to i in D minor:
Corelli Concerto Grosso op. 6 no. 8