Heraldry, also known as Armoury, developed in 12th Century Europe as a way of identifying individual knights and nobles in battles, tournaments and pageants. The need to identify individuals and their retinues easily and at a distance meant that large, bold shapes and bright colours were used.

Originally, particular colours and designs would have been carried on banners, shields and surcoats worn over armour - hence the predominance of the shield in Heraldry and the very term "Coat of Arms". There are specific and well-defined rules by Arms are designed - there had to be, since the aim was to be able to identify anyone by "reading" the Arms. These rules are numerous but straightforward and rigorous. Together, they are known as the "blazon" (description) of the Arms.

In addition to the standard shield, there are often coronets (caps of rank) such as the helmet, supporters (often fabulous animals) to either side and a motto usually in an "escrol". Sometimes the shield stands upon a Ground or Compartment, typically made to represent a grassy mound or seascape and often, in Scotland, with the Clan or family plant badge depicted.

The helmet will often bear mantling (the tattered cloth on the top and sides of the shield), a wreath (on top of the helmet) and a crest (like the hand holding a crescent in the Durie Arms). There may be other badges of rank such as an Earl's cloak or an ecclesiastical symbol.

By their very nature, Arms can only belong to one person at a time. There is no such thing as "family Arms" or a "Clan crest" and no-one has the right to display Arms without having these verified and registered ("matriculated") by the appropriate authority. In Scotland this is the Lord Lyon King of Arms, essentially the senior judge in the specific court of Heraldry, and with quasi-regal powers.

Not only is it illegal to bear Arms that have not been specifically awarded by the Lord Lyon, it is also illegal to invent Arms or to use existing or invented Arms.

No two people can bear the same Arms. However, Arms are heritable property and are inherited by the petitioner's heir, normally his eldest son, and by his eldest son in turn. The father's Arms can be matriculated in favour of the son upon death and succession, but this is not automatic - the matriculation process must be gone through before there is "achievement" of Arms. Younger children inherit only a right to matriculate the Arms with a slight difference added, and they must petition separately for this to be done.

On the other hand, Scotland Heraldry is such that almost all Arms granted to persons of a certain name are based on the Arms of the chief of that name, even when there is distant or no blood relationship. Scottish Arms are therefore more traditional and use mediaeval charges and patterns of charges, unlike English and other Arms.

Scottish heraldry differs in many other respects from English and other European heraldries. 

A useful link: http://www.heraldry-scotland.co.uk