I suppose the principle of correspondences can be summed up in the words of the abbreviated Emerald Tablet of Hermes Trismegistus:
As above, so below
Correspondences are symbols which magically resonate together. They can be ideas, physical objects, colours, crystals, shapes, plants or animals, but they are all used to add power, success and effectiveness to spells and magical workings.
Once you start doing any magical working, ritual or spell, you will naturally begin to use correspondences without thinking about it, and without even realising you are doing it.
For example a simple spell for good luck:
On a Wednesday with the moon waxing, mark an orange candle with the symbol of Mercury. Light the candle and put near it a marker you will use in Bingo and say:
Lady Luck, Lord of Chance,
Make the Bingo numbers dance
Help me win and fill my need,
Make numbers fill my card with speed!
Let the candle burn down, then pick up the marker pen and say:
My Lucky Pen will help me win!
Obviously take the pen with you to Bingo and use it to mark your card.
The above spell makes use of quite a few different correspondences. First the day of the week, Wednesday is sacred to the god Mercury/Hermes who devised the first gambling game, bones or dice.
The Moon waxing is used to increase anything as the shape of the Moon can be seen as waxing or growing towards full. In this case we are 'growing' our Good Luck.
Orange is one of the colours sacred to Mercury and this is emphasised by marking the candle with the symbol of Mercury too.
The marker pen you put by the candle is being infused with the energies of Good Luck which you are gathering with your spell, and becomes a lucky talisman for you to take with you to Bingo.
Very often the sheer number of correspondences that an experienced magician or Witch incorporates into their rituals can be seen as overwhelming to a beginner. There are books written exclusively on this subject including '777' by Aleister Crowley.
But really there are a few major ones which can easily be learnt: the energies of the phases of the moon; the energies of the days of the week; and the significance of colours.
Many other correspondences are based on these few simple basic correspondences. For examples the correspondences of plants and trees is based on the Doctrine of Signatures which says that each plant contains indications of what it might be useful for, either in its colour, shape of leaf or root etc. This in turn leads to each plant being assigned and astrological planet, which again has the same correspondences as the days of the week.
The significance of colours often also indicates what a particular gem stone could be used for. So pink stones are sacred to Venus, yellow ones to the Sun and so on.
One other point about correspondences is that they can also depend on what you feel about a colour or object. And upon your cultural background too. For example to the Chinese red is the colour of wealth and good luck, so you could use a red candle in the above spell, and instead of marking it with the symbol of Mercury you could mark the candle 88 as the number eight is seen as being very similar to the Chinese symbol for happiness and luck, so 88 is the equivalent to the Chinese symbol Double Good Luck, which is a very potent and fortunate symbol to use.
Have a look at our Fact Sheets number 2 Candle and Colour Magic and 49 Planetary Magic, these will give you the basics of correspondences and help you get on your way to effective, practical magic