A war-cry (French: cri d'armes, cri de guerre, cri) is what its name suggests, something that one (or one's followers) shouted as one threw oneself into the fray. Very often, the war-cry was simply the family name. I once heard that the pronunciation of the Scottish name Home was changed to Hume after disastrous results during a battle when the war-cry "Home! Home!" was shouted and soldiers all headed home. (Si non e vero...).
In English heraldry, War-cries, like mottoes are unregulated, not necessarily part of the grant of arms, and liable to change at will. In Scottish heraldry, the motto is registered, and so is the war-cry or "slogan" (slogorn, which literally means war-cry in Gaelic) which is used by the chief of the clan only and appears above the crest (in which case the motto, if any, appears below the achievement instead of above as usual; see Innes of Learney, Scots Heraldry, p.39).
The war-cry is very old, in some cases it probably precedes heraldry, some cries are cited in gestae of the 12th c. The motto (Italian for "word") really only appears in the 14th c., and becomes fashionable in the 15th and 16th c. (in parallel with the development of badges; a combination motto + badge forms an impresa). Its origin and purpose is really different from the war-cry: the motto is originally a personal choice (German Wahlsprüche) designed to express some particular quality, intention or event. The war-cry is more feudal in nature, used by the chief of the house and sometimes shared by various vassals, or modified (see, for example, the "mark of cadency" on the Capetian dukes of Burgundy's war-cry).
Of course, in some cases war-cries became mottoes: thus is the case, probably, for Hamilton (Through), Douglas (Forward), Fiztgerald duke of Leinster (Crom a boo = I will burn). The most famous instance is the English royal motto: Dieu et mon droit, supposedly the war-cry used at the battle of Crécy in 1346 (God and my right, i.e., to the throne of France).
War-cries are often the family name repeated, or else (example of Surreol, Vendome below) the name of the liege lord. Many war-cries are religious invocations (e.g. Normandy, Limoges, Anjou), rallying cries (Flanders), cries of encouragement (Tournon), cries intended to strike fear (Bar).
Fox-Davies claims it is exclusive to French and British heraldry, though I see mention of a war-cry for the German Emperor: "dexter and sinister" (i.e. strike left and right).
Here are a few examples of war-cries: