Weapon Foundries are the arms manufacturing corporations of the City. The names of most weapon foundries were introduced in the Destiny base game, but they play a much bigger role beginning in The Taken King. At least one new weapon foundry, Veist, is introduced in Destiny 2.
Each foundry has their own unique style and feel, emphasizing different design philosophies. In Destiny, legendary foundry weapons from Häkke, SUROS, or Omolon can be purchased with randomized perks from Banshee-44 on Armsday. Cassoid, Crux/Lomar, Daito, Nadir, and Tex Mechanica are responsible for producing several exotic weapons.
Weapons from different weapon foundries tend to handle slightly different, whether it having different fire modes available on certain weapons, prioritizing different stats, or having certain perks available. See each weapon foundry page for more details.
According to Commander Zavala, the weapon foundries began from the traditions of tribal scavengers who repaired and restored the ancient technology of the Golden Age. These gunsmiths eventually became the large corporations of the City, but they became a potential threat to the City's common good (likely referring to involvement in the Faction Wars). The City now monitors the size of the weapon foundries, and the surviving foundries continue to use new discoveries to improve the Guardian arsenal.