This article is from the rec.heraldry FAQ, by Gordon Findlay (email@example.com) and Francois Velde (firstname.lastname@example.org) with numerous contributions by others.
Without knowing the circumstances of the original grant, it is difficult
to say whether a coat means anything at all, except that someone
(grantee or herald) liked the design.
Some arms ("canting" arms) contain a charge whose name is related to the
surname of the bearer (e.g. de Trumpington: Azure, crusily, two trumpets
pileways Or). This can be taken to the extent of becoming a rebus
puzzle -- the Borough of Congleton bears Sable, on water in base
barry-wavy azure and argent, on a tun between two conger eels argent, a
lion statant-guardant Or, which decodes to Conger-Leo-Tun.
In the Middle Ages, bestiaries, popular tales and folklore contributed
greatly to the association of specific animals with specific characteristics
or virtues, some of which persist to this day (owls are wise, elephants
have memory, etc). It is quite possible, for any given coat, that the
original bearer chose an animal with such associations in mind.
Often a coat will contain charges alluding to the original grantee's
career or interests; for example medieval merchants and guildsmen often
included the tools of their trade. These may become less appropriate as
the coat is passed down through the generations, or their significance
is forgotten. Quite elaborate schemes can be developed: a former
Governor General of New Zealand has a coat based on the theme "a cat
among the pigeons", which is apparently how she sees her career.
Some charges were taken from the arms of a bearer's feudal lord or
protector as a mark of loyalty. For example, the Maltese cross in the
arms of several towns in Switzerland is a reference to the Knights of
Malta, who were once sovereign in that area. The frequency with which
the bar, a type of fish, appears in coats of arms of the former duchy
of Bar in Eastern France can only be explained in this way. Also, imperial
eagles which appear in many Italian coats were originally meant as a sign
of allegiance to the Imperial party in the conflicts which tore medieval Italy.