At skill level and character level 40, Blacksmiths could choose to specialize as an Armorsmith or a Weaponsmith.
Weaponsmiths could later further specialize into hammer, axe or swordsmiths. One of the poorly executed ideas from Blizzard as far as professions go, it was tied to the Mithril Order pseudo-faction and peaked in Burning Crusade which added weapon sub-specializations.
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The related recipes in Wrath of the Lich King became largely irrelevant and the feature was completely removed in Cataclysm for the most part. Becoming an Artisan Armorsmith gave you access to a number of new, powerful armor recipes.
Specializing in Artisan Weaponsmithing granted access to devastating weapons not normally available. Base prerequisites were character level 40 and a blacksmithing skill of Choosing to specialize as an armorsmith would have precluded you from specializing in weaponsmithing and vice versa.
Horde players would first speak with Krathok Moltenfistto start the quest chain for becoming an Armorsmith or a Weaponsmith while Alliance players would speak with Myolor Sunderfury to start the Alliance versions of the same quests.
They would both direct you to the appropriate quest giver behind them. Blacksmithing specialties could be changed at the trainers for a cost.