Forged weapons are made from metals, woods and other resources from the environment. Beastcraft weapons are made out of animal-derived materials.
Processed weapons are a mix of the two materials that has then been reinforced with alchemy. An example is Carmitization, which turns organic material a dark red and makes it stronger, lighter and more flexible.
Alchemica are high-class Beastcraft weapons that utilize prana to replicate a living risk beasts special abilities and traits. In older character profiles, these are named Alchemic weapons.
Machina are high-class Forged weapons that utilize prana to perform unique functions. In older character profiles, these are named Mechanical weapons.
Both types of prana-powered weapons use an Artificial Chakra, a horn-like extension, to absorb prana from the air.
these classifications apply to non-weapons as well, of course. A hammer made from beast iron and wood is a forged tool, while needles made from beasts are beastcrafts.