In his brilliant book 'One Two Three Four', Craig Brown says that they represent the four elements:
John is fire
Paul is water
George is air
Ringo is earth
I would disagree with this. Paul is air and George is water. This is backed up by their star signs, Paul Gemini (air), George Pisces (water). Still waters run deep, that's George!
In fact, John is actually Libra (air) and Ringo Cancer (water) so there was no fire sign in the group. Although I agree John was fiery and Ringo earthy, so their personal horoscopes may show those elements on the ascendant or something. I think that can change things.