The idea of a coat of arms first developed in the 12th century or so, to aid in battle. Without uniforms or other distinguishing features it was often difficult to tell friend from foe in the heat of battle, and so coats of arms were designed as identification marks to indicate who was who and as banners to indicate where troops should go. The concept then spread through the population, with even those who would never go into battle taking on a coat of arms and passing it onto their descendent. Thus began the concept of "family" coats of arms.
While in America coats of arms are treated rather liberally, with anyone able to buy their apparent "family crest" (the name being a misnomer - a crest is an aspect of the full form of coat of arms) from what are known as "bucket shops", in Europe things are much more strict. Many governments have rulings on the books that one cannot display a coat of arms without being able to actually trace your lineage to the family that bears it. In Scotland in particular, the coat of arms can only be borne by one person at a time - sons of the family instead bear modified versions indicating their place in the family tree, with the first son removing this mark when the coat of arms is passed on. Other countries are less strict, but still follow rough rules of inheritance and patterning.
One interesting aspect of heraldry is the language involved with it, which is similar to pure English but patterned with French and technical terms. The description of a coat of arms often comes off as incomprehensible to the casual reader - a shield that depicts a simple blue field with a white stripe down the center would be named as "Azure, a pale argent", for example. More complex shields of course demand much more complex vocabulary. The arms of Hungary under the Austrian-Hungarian empire provide a clear example of this:
Quarterly, I azure, three lions' heads affrontés crowned Or (for Dalmatia); II chequy gules and argent (for Croatia); III azure, a river in fess gules bordered argent thereupon a marten proper beneath a six-pointed star Or (for Slavonia); IV per fess azure and Or, overall a bar gules in the chief a demi-eagle sable displayed addextré of the sun in splendour and senestré of a crescent argent in the base seven towers three and four gules (for Transylvania); enté en point gules, a double-headed eagle proper on a peninsula vert holding a vase pouring water into the sea argent beneath a crown proper with bands azure (for Fiume); overall an escutcheon barry of eight gules and argent impaling gules, on a mount vert a crown Or issuant therefrom a double cross argent (for Hungary).
Obviously, the language of the blazon is somewhat difficult to understand. However, a few simple terms can be learned. The right of the shield for the bearer (the left for the viewer) is dexter, while the left is sinister. Colors names are derived from Old French - red is gules, black is sable, blue is azure, and so on. Yellow and white are defined as being or and argent, the colors of gold and silver, and are the two metals of heraldry. The shield itself is the field, and it can be divided either simply ('party per pale' means the field has a division down the middle, for example) or in more complex patterns ('party per saltire' means the field is divided by a diagonal cross; 'barry' means a shield is covered in alternating bars). Ordinaries may also be present, simple geometric figures of one color or another. The dramatic lions and crosses and eagles that make heraldry are known as 'charges'. Animal charges are particularly complex, possessing various 'attitudes' - the lion rampant is perhaps the best known, but one may also have an eagle 'displayed', a lion 'passant guardant', or a horse 'courant', among many others. In a full coat of arms, many figures surround the shield, including animal 'supporters' to bear it, a symbolic 'crest' atop it, a 'helm' and 'mantling', and far more, giving more detail and symbolism to the coat of arms overall.
While heraldry is now a lesser-known art, many of its elements are still reflected - for example, many national flags (such as the Union Jack of Britain) follow what is known as the 'rule of tincture', which holds that a color cannot touch another color and a metal cannot touch another metal. Look at the flag and notice that the white 'outside cross' divides the red from the blue effectively and neatly. Heraldry may be fading from fashion and notice for all but the nobility, but its elements still can be seen in the design of flags, logos, symbols, and much much more.