Coats of arms are often portrayed and displayed as simple shields, but in heraldry they are more properly displayed as a full heraldic achievement. In an achievement, the shield is held upright by two supporters, often human, though sometimes animal. The supporters and the shield stand on a mound, usually shown as being covered in grass, and sometimes with a family motto underneath. Above the shield will be a helmet denoting the noble rank of the bearer of arms, and above that will be a crest, often an animal. The crest will often be on a wreath atop the helmet, from which will flow long, flowing furs, called mantling. This mantling is a stylised representation of furs which were intended to cool the head of a knight in full armour.
Returning to the shield itself, also known as an escutcheon, it should be noted that the symbols shown there are described as charges on a field. These charges often take the form of a basic set of shapes known as the ordinaries. These include shapes such as the chevron, fess, bend, pale, and saltire. Many other charges appear on shields, however. These include animals such as lions or bears, plants such as trees, or indeed any other object you might imagine.
The colours used in heraldry, known as tinctures, all have peculiar names. There are two metals, or and argent, or gold and silver, and normally shown as yellow and white. Furs include ermine and sable (black), and colours include gules (red), azure (blue), and vert (green). The Rule of Tincture states that no colour may be placed on a colour, no metal on a metal, and no fur on a fur, with certain exceptions. It should be noted that colours can be placed side-by-side - the rule is merely that, for example, a red charge cannot be placed on a green field.
Part of the charm of many coats of arms derives from the process of quartering. Where a man marries an heiress who is also entitled to bear arms, their children will have their arms quartered to show both their parents' family devices. As following generations marry, quarterings can become progressively more complicated, although modern heralds tend to prefer simpler arms, while showing as many illustrious forebears as possible.
A further complication when looking at shields is introduced by the process of differencing. This system arose from the need to differentiate between the arms of a second son and those of his father. While the eldest son and heir would inherit his father's arms, his second son would bear an arms differenced by means of a bordure (border), charge, or some other method.
This concludes my brief look at heraldry and the evolution of arms. More detailed study will reap its own rewards, and I especially recommend Gritchka's fine heraldry nodes here on E2.
Discovering Heraldry, Jacqueline Fearn, Shire Publications Ltd, 1992
Heraldry in England, Anthony Wagner, Penguin Books, 1953
Observer's Book of Heraldry, Charles Mackinnon of Dunakin, Frederick Warne & Co Ltd, 1980
Encyclopaedia Britannica, Inc, 1994-2000