It is fruitless to try to determine "the meaning" of a specific coat of arms.
Coats of arms were, for the most part, freely adopted by individuals at some point in the Middle Ages or later. In almost all cases, it is impossible to say what meaning, if any, they saw in the arms they adopted. Even in the case of grants of arms, it is usually not known who designed the arms (the recipient or the royal herald) and what he had in mind.
There are exceptions, of course.
For example, there are cases where an anecdote, true or legendary, is attached to the origin of a coat of arms (see examples below). Also, for the most prestigious and important coats of arms, such as those of kingdoms, legends built up over their origins, and symbolic meaning was attached to them as well. In some cases, modern scientific research has been able to provide some guesses as to the origins of particular designs (see for example the case of the fleur-de-lys in France).
One broad category of arms for which the meaning is easy to guess is that of canting arms (armes parlantes): those are arms for which the blazon (the verbal description) offers a pun on the name of the family. Examples are extremely numerous, and in many cases some unusual or bizarre charges have come into heraldry solely for the purposes of a pun. I have collected some examples of canting arms.
Another category of examples concerns augmentations, that is, additions to a coat of arms either granted by a sovereign or adopted by a bearer to publicize his allegiance to an individual or cause. Many Italian coats of arms have a label gules or even a chief azure with fleurs-de-lys or and a label gules to promote their faithfulness to the Angevin kings of Naples. During the conflict between the German Emperor and the Pope, a number of Italian families received or adopted an eagle sable as mark of allegiance to the Emperor. Later, during the 15th and 16th centuries when France was involved in Italian politics, the arms of France appeared on Italian coats of arms (a famous example being the Medici).
It is true that some charges have some symbolic meaning attached to them, though the meaning is by no means universal nor unambiguous. Medieval men were fond of legends attributing peculiar behavior to animals, and relating those examples of behavior to human virtues or vices: thus, the animals became emblems of such virtue or vice. The eagle's presumed ability to stare at the sun without becoming blind became a source of metaphors. The pelican was supposed to open its breast with its beak to feed its young in times of distress, and the bird thus became a symbol of generosity or parental devotion (and also of the Redemption, which is why it is often shown above the cross in 14th c. depictions of the Crucifixion). As such, it is almost always represented in the act of so feeding its young in heraldry. As an other example, the bee, because of its behavior, was long seen as a symbol of industriousness, and thus became a favorite of Englishmen enriched by the Industrial Revolution. The general phenomenon of medieval bestiaries is discussed below.
However, even when it is known that a certain animal is often associated with a certain trait or abstract notion, it does not mean that any coat of arms with such a charge was intended to recall that trait or bring to mind that notion. It is thus pointless to try to decipher a coat of arms when one has no other specific information about the origins of the arms.
Italy was a constant battleground in the Middle Ages: the struggle between the Pope and the Emperor, which extended into the long-lasting Guelf/Ghibelline conflict, was later followed by the wars between France and Spain over control of the Peninsula. As a result, many Italian families have adopted or received augmentations as indication of the side they had chosen.
The terms of Guelfi and Ghibellini come from a dispute between two German families, the Hohenstaufen (lords of Weibelingen) and the Welfen, over the Imperial throne, between 1138 and 1234. This dispute ended in Germany but its repercussions continued in Italy. The two terms appeared there in 1218, and came to mean partisans of Italian freedom from foreign intervention, support of the Pope, and democratic institutions (Guelfi) versus partisans of the Emperor's participation in Italian affairs and strong central authority (Ghibellini). The Angevin dynasty in Naples (1266-1435) was the Pope's ally against the Emperor, hence the chief of Anjou, which recalls the arms of this junior branch of France. In practice, the terms were used to designate long-standing rivalries, and Pisa was Ghibelline because Florence was Guelf; and "per fess" meant Guelf because "per pale" meant Ghibelline! Florence was a long-standing Guelf city, and the arms of the Guelf party were an eagle preying on a wyvern and surmounted by a fleur-de-lys flory.
Here are some charges which are characteristic (taken from Piero Marchi: I Blasoni delle Famiglie Toscane; Roma, 1992).
Under construction.
One of the most popular books (or type of books) in the Middle Ages was the bestiary, a collection of 30 to 60 descriptions of animals (and often plants and stones) with their characteristics and habits, and the symbolic significance of those characteristics.
The common origin of these collections is the Physiologus, a pagan text written sometime in the 2nd century CE in Alexandria (Egypt) or perhaps Syria; much of its contents was derived from the Antique world's general store of animal legends and stories, as recounted in Herodotus, Aristotle, Pliny, etc. Sometime in the 3d or 4th century AD a Christian version was written. The work was not driven by an interest in the animal world itself, but rather by the idea that the physical world should be read like a book, for signs of God's wisdom. Thus, animal behavior was interpreted as symbolic of virtues or vices which are condoned or rebuked by Christianity, or even as symbolic of crucial theological points (the pelican's story interpreted as the Redemption).
Latin versions of the Physiologus abounded in the 11th and 12th c., and were circulated widely. At the same time, new compilations were made, in Latin or in the local vernacular (French, English, Italian), and beautifully illustrated versions were made. The peak of the bestiary vogue seems to be around the 13th century, which happens to be the formative period of heraldry. The vogue declined in the 15th c. and bestiaries were seldom produced after 1500, although some of the characterizations of animals survived independently.
Given how widespread these symbolic readings of animals, plants and stones were, it is not unreasonable to look to Medieval bestiaries for clues as to how charges were seen and what kind of connotations were attached to them, particularly for more uncommon animals.
See a beautiful digital edition of the Aberdeen bestiary (12th c.).
In the late 14th century, a fad developped across all Europe, probably originating from Italy, for a peculiar kind of personal emblem. It usually consisted of a badge (either a simple object or animal, sometimes almost a little scene or allegory) and a motto (literally, "word" in Italian). The combination image + word was called an impresa.
In English and French armories, one finds such imprese quite commonly. They were adopted by an individual for a period of time or for his whole life. The choices of badges could bear a relation to the individual's family heraldry, or not.
As a Renaissance equivalent to bestiaries, we have popular books of emblems, providing long lists of themes and their symbolic representation. One example is available on-line: it is Alciato's Emblematum Liber (1531; the 1536 French edition is also available at the University of Glasgow). It begins with the arms of the duke of Milan, to whom the book is dedicated.
For some coats of arms, anecdotes are told about their origins. Whether true or not (and they are probably more often not true), they are entertaining (the arms are authentic in each case, and born as described by each family).
Source: JM = Jougla de Morenas.