All the information in this post is based upon my own observations and I borrowed a few things from the improved boss guide. Some of it may be incomplete or flat-out wrong.
Shogun’s Castle bosses
1st floor
Boss name: Saigo
Weaknesses: None
2nd floor
Boss name: Nobunaga
Weaknesses: Mechanic
3rd floor
Boss name: Grand General Shinoda
Weaknesses: None
4th floor
Boss name: Shogun
Weaknesses: Mechanic
War Factory bosses
1st floor

Boss name: Mr.Wong
Health: 135070
Weaknesses: No info
Strengths: No info
2nd floor

Boss name: Firecracker
Health: 175000
Weaknesses: Earth, Water
Strengths: Beast, Ice, Plant
3rd floor

Boss name: Whalthzer
Health: 302200
Weaknesses: No info
Strengths: No info
4th floor

Boss name: Lost Crusader
Health: 205500
Weaknesses: Earth, Water
Strengths: Beast, Ice, Plant
5th floor

Boss name: Envenenador
Health: 345800
Weaknesses: No info
Strengths: No info
6th floor

Boss name: Marauder
Health: 480600
Weaknesses: Earth, Fire, Light
Strengths: Mechanic, Plant
7th floor

Boss name: Bison Walker
Health: 445000
Weaknesses: Earth, Fire, Light
Strengths: Mechanic, Plant
8th floor

Boss name: Duke Vincent
Health: 389000
Weaknesses: Earth, Water
Strengths: Beast, Ice, Plant
9th floor

Boss name: Great Thorlock
Health: 467000
Weaknesses: Earth, Fire, Light
Strengths: Mechanic, Plant
10th floor

Boss name: Fire Lord
Health: 686000
Weaknesses: Earth, Water
Strengths: Beast, Ice, Plant
- images of Great Thorlock and Fire Lord are reduced to a more cute and cuddly size
List of bosses by element
I’m pretty sure each boss has its own element which determines what it is weak or strong against. See the table in the next section for the strengths and weaknesses.
Water: Dark Deep Sea Predator
Ice: Viscount Vilhelm, Ice Hearted Prince
Fire: Grand Lava Keeper, Firecracker (WF), Lost crusader (WF), Duke Vincent (WF), Fire Lord (WF)
Light: n/a
Darkness: Grand Montero, Angel of Destruction, Fuerte Ira, Void
Plant: Buldorados
Mechanic: Battle Conqueror
Earth: Warlord Zirhon, Prince Dulahan
Beast: Toxico Spider, Marauder (WF), Bison walker (WF), Great Thorlock (WF), Guardian Beast
Flying: Maestro
Lightning: Pirate Captain
No element: Banditor, Pontero, Destroyer, Khaz’ar, Mr Wong (WF), Whalthzer (WF), Envenador (WF)
Elemental strengths and weaknesses

+: The element to the left is strong against the element above
-: The element to the left is weak against the element above
.: No relation
There’s a question mark for the relation between light and plant because as of yet there are no bosses of either element.
Update: There doesn’t seem to be any relation between light and plant.
Damage formula
The damage formula for each strike is:
( ( base attack x skill attack x elemental multiplier ) – enemy defense ) [ x critical ]
Each elemental strength or weakness is a multiplier in itself, so combining elements on your gear that an enemy is weak against will make you do more damage. For example, with +1 water and +1 earth you will do more damage against fire bosses than just +2 water or +2 earth. To calculate the total elemental multiplier, you have to multiply all the strengths and weaknesses.
For each strength: 1 + ( 0.5 x elemental bonus )
For each weakness: 1 / ( 1 + ( 0.5 x elemental bonus ) )
Multiply all these values, and you will have the elemental multiplier.
Example: A critical strike of the punishment skill (level 5) with +2 water and +1 earth, a base attack of 1176 and x7 crit against a fire boss with no defense would do
( ( 1176 × 1.9 x ( 1 + ( 0.5 × 2 ) ) x ( 1 + ( 0.5 × 1 ) ) ) – 0 ) x 7 = 46922.4 damage