The knowledge of
arms, the
emblems used in
mediaeval warfare: the
shields,
banners,
helmets, and
liveries of the King and his
lords and their
followers.
From this job of the herald as a marshal and messenger on the battlefield arose the ancillary duties of confirming the genealogy of those entitled to bear arms, and organizing displays, such as parades, tournaments, and coronations, at which arms were prominent.
A narrower field than heraldry is armory, which is specifically the study of the emblems on the arms. Armory is the central part of heraldry. (Note. Only the spelling armory is used in this sense: as opposed to an armoury or armory, a place where arms are stored.)
Warriors have always used emblems, and the shield is a handy place for them: the Greek poets contain a number of descriptions of them. But the coat of arms as a hereditary indication of one's status in society is a development of the Middle Ages. By the 1300s the granting of arms was the prerogative of the King. Heralds kept records of them and advised the King about which were genuine and rightfully borne. (In what follows it should be obvious when I am specifically talking about England: the general principles of Scottish and Continental heraldry are the same.)
The trial of Scrope v. Grosvenor established that no more than one person was entitled to a particular armorial bearing. This was held in the Court of Chivalry, which is a court of law that still exists. Dormant from 1737, it was revived in 1954 when the city of Manchester prosecuted someone who had used the city arms without authority.
It is worth emphasizing that a coat of arms belongs, at one time, to one person. It is not shared as such by a family, and certainly not by a mere surname. All male descendants of an arms-bearer can themselves bear arms. Younger sons are allowed to use the family arms with an additional emblem: these distinguishing emblems are called marks of cadency. People whose name is Jackson are no more allowed to use the 'Jackson' arms than they are allowed to take cars belong to other Jacksons and drive them away. The difference between arms and a car or a gold watch is that arms descend in the family. A Sir William Jackson can bequeath his gold watch to his doctor, but he cannot change who is to inherit his arms. See heir for details of how these can be inherited. Sites or companies claiming to show 'family' arms are completely bogus.
Heralds were organized into a college or guild called the College of Arms by Richard III in 1485. Heralds originally served other lords as well as the king, and tend to have territorial designations: as Richmond Herald of Arms; or from the order of knighthood they look after. A junior herald is called a pursuivant.
The senior heralds have the peculiar title of King of Arms (or King at Arms). I have no idea why. It does not seem to derive from a title along the lines of "king's adviser". The chief of them is Garter King of Arms, whose remit is the Order of the Garter. The other kings of arms are territorial: Norroy in England north of the River Trent, Clarenceux in the south of England, and Ulster King of Arms in Ireland. (Ulster was for all Ireland, but now that only the north is part of Britain, Norroy and Ulster are always combined in the same person.) In Scotland the Lord Lyon King of Arms is the supreme arbiter of arms and chiefship, and the Scottish equivalent of the Court of Chivalry is called the Court of Lord Lyon.
The central article of an armorial bearing is of course the shield. A coat of arms has to have a shield, if nothing else. Often it has all sorts of gubbins, fal-lals, and doodahs all around it: the supporters holding it up, the helmet above it, the crest on top of that, and a few others. The whole thing, with the shield in the middle and its trappings all round it, is called the coat of arms or the achievement. (It is not called a crest.)
Heraldry has a wonderful language all of its own, in which a shield divided horizontally red above blue with a white star on it would be described as: Per fess Gules and Azure a Mullet Argent. This formal description is called blazon.