Sorry for the late response. Information on the extra properties (e.g. fire) would be lovely.
As for weapon types, well, that seems more of a bookcase read or Explorer Society conversation.
Ah, alright. Here's some relatively early weapons that have special properties [as most don't]:
Breaker [Dagger}: Has an 80% chance of lowering the target's DEF.
Air [Dagger]: Deals air element damage [in addition to the dagger's usual phys/slash damage]
Luck [Claw]: Lowers the target'sEVA.
Char/Chill/Ground [all whips]: Does Fire/Ice/Earth damage [in addition to the whip's usual phys/slash damage].
Star [One each of sword, rod, and bow] Does Lux damage [in addition to the weapon type usual damage types].
Water [Hammer]: Deals Ice damage [hopefully you get the idea].
Dread [sword]: Has a chance of dealing a deathblow [the target's HP in damage, affected by deathblow resistance] to its target.
Catch [spear]: 'Surveys' [steals] an item from its target with its normal attacks.
There's a bunch more weapons with special properties, but a lot of those come later in the game, so hopefully this is a good enough overview of things.