So what does all this mean?
The circle works using the interval of a 5th - the distance between the open strings on your instrument eg. D to A. Or on a piano, counting seven semitones (half steps) up from a note (seven notes including black keys to the right).
This means that every step to the right of the circle moves up a 5th. Every step left moves down a 5th.
The starting point of the circle is the key C major. This key (or scale) has no sharps or flats. A step to the right gives us G major, which has one sharp. Each step to the right gives the new key (a 5th up) which has one extra sharp.
This also works in the reverse direction (from our starting point of C major) with flats, moving down a 5th each time.
The outside circle of the diagram shows you the major scale keys. However every key signature is shared by two types of scale - the major and minor. The major and minor scales that share a key signature are often referred to as 'relatives'. The relative scale of C major is A minor. The inner circle therefore shows you the minor scale keys, starting with A minor (with no sharps or flats). It then works in the exact same way as the outer circle.
This is the basics of the circle. In Circle of 5ths - part 2 I go into further detail about the order of key signatures, and 'relative' scales.