Heraldry (the “noble science”) is the study, profession, or art of devising and granting arms and ruling on questions of heraldic protocol.
A coat of arms is a design used by a person or group of people (a nation or town, for example) in a variety of ways. Their original purpose was to idenify knights and soldiers in battle, but as time went on, heraldry become more and more confined to the graphic instead of practical and military realm.
There are rules used in heraldry that regulate the way arms are designed, described, and used. These rules are not absolute, and are followed to varying degrees across Europe and the world. Basically, there is a distinction between heraldic rules and heraldic customs. The former are adhered to everywhere, universally, while the latter are country-specific or region-specific and add a local flavour to the world of heraldry and blasons. Thus, Anglo-French heraldry has some art and blasoning customs different to those of Scandinavian heraldry, Italian heraldry, German-speaking and Slavic-speaking central European heraldry, etc.
AH.com Heraldry - Some parts of AH.com had or have their own semi-jokey heraldry too, of course.