Here is a highly useful mnemonic illustration of the relationship between chords in different keys, known as the "Chord Clock":
The clock illustrates the six chords used for the key of "C", namely C, F and G7th, Dm, Am and Em, which are the only chords needed to accompany a song in C on the piano or guitar.
The clock enables one to transpose a song from one key to another very quickly. For example, to convert the chords of a song in the key of C into the chords required to play it in the key of G, one just "moves" the position of the clock one space to the right, so as to find the alternative chords to play. For example:
from Key of C: C-G-C-Am F-G7-C
Note that the clockwise order of chords is the same throughout (FCGDAEB then sharped F#,C#,G# etc) for the major (outer circle) chords and also for their relative minor (inner circle) chords. This is because of the harmonic relationship between the actual notes that make up each chord - the root note and its relative third, fifth and octave notes (for example C,E,G and octave C). In the key of C, the G note is a fifth (or five notes) higher than the root note C. Then taking the fifth (G) as the basis for the next key (ie. the key of G), the D note is the corresponding fifth note up from G in that new scale, and so on.
The order of chords (FCGDAEB etc) is best remembered by using the same mnemonic that identifies musical key signatures on sheet music, namely:
Mnemonics Guide Page ©2000 An EUdesign site