What are Armaments on weapons?

[K.I.L.E.R]

Things like Dark Magic Reversal.

Special attacks proc off default weapon attacks or does it have to be from that weapon itself?

[Landale]

Armament and Special Attack are two different things.
Special Attack is a special attribute outside the normal six, which varies based on what it is, and deals more damage against a type of enemy. Lightning Blade, for example is strong against aquatic enemies. Dark Magic Reversal, on the other hand, seems to work best against constructs like the Pins.
Armaments, on the other hand, are a cost of use. These will always drain out HP or DP, and if you don't have enough will make the character faint.

Some weapons can have both though, like the Spike Shield katar type weapon having Armament: HP Cost and Sp. Atk: Bitterner