The Circle of Fifths is a wonderful, visual representation of the relationships between the major keys, the minor keys, and our twelve key signatures. As a tool, we can easily figure out the number of sharps or flats in a given key signature and what those sharps and flats should be. For instance, if someone tells us to play in the key of E, we can see that the major key of E is at the 4' o-clock position which means it should have 4 sharps in the key signature. The first sharp is always F#. The next sharps follow the letter sequence on the outside of the circle in a clockwise direction. So, in the key of E, the four sharps would be F#, C#, G#, and D#. Similarly, if we're asked to play in the key of Eb, we know from the chart Eb is the third spot down on the left of the chart going counter-clockwise. Therefore, that key signature must have three flats. The first flat is always Bb. The next two flats then after Bb following the letters counter-clockwise on the outside of the circle are Eb and Ab. The key of Eb then has three flats in the key signature: Bb, Eb, and Ab!
We can also use the chart to quickly identify the sub-dominant and dominant chords associated with a particular root. For example, in the key of C, the sub-dominant chord (or IV chord) is one to the left on the circle: F major. The dominant chord (or V chord) is one to the right on the circle: G major. In the key of Eb, the sub-dominant chord (or IV chord) is one spot counter-clockwise on the circle: Ab major. The dominant chord (or V chord) is one spot clockwise on the circle: Bb major. In a typical, major chord progression, we would then hear I-IV-V-I or Eb major (the root), followed by the Ab chord (the sub-dominant or IV chord), next the Bb major chord (the dominant or V chord), and a return to the root or tonic (I-chord) Eb major. Cool, huh?
The bottom three key signatures are our "enharmonic" keys. The smaller, inner circle (in lowercase letters) shows us the relative minors of the major keys.
Below are links to a downloadable image for printing (courtesy of LinkWare Graphics) and to some other websites with more detail about the circle of fifths.