Damage Types
Each damage source has up to four types, consisting of the Major damage type (Physical or Elemental), the Damage Subtype, the Major Item type, the Item subtype. Specific bonus types may also apply. Any inscriptions or effects which affect one of the damage source's types will apply to it.
The total damage multiplier for the damage source can then be found by adding all of the bonuses which affect it. In this way, each of the damage source's types can be thought of as a "modifier slot": When determining how damage modifiers will apply, just determine the modifiers for each slot, and add them together.
Slots 1 and 2: Major Damage Type and Damage Subtype
The Major damage types are Physical and Elemental. Impact and Acid are Physical damage, and Electric, Fire, and Ice are Elemental damage.
The Damage Subtypes are Impact, Acid, Electric, Fire, and Ice. Bonuses to these apply in addition to the general elemental/physical bonus, as they are in separate "type slots"
Slots 3 and 4: Major Item Type and Item Subtype
The Major item Types are Weapons and Gear.
The Item Subtypes for Gear are [Q/LB] and [E/RB]. Item Subtypes for weapons are quite simply the weapon category: Assault Rifle, Marksman Rifle, Autocannon, etc.
Damage which does not come from an item will not have a Major Item type or Item Subtype, but will usually have an associated bonus type which affects it.
Bonus Slots
Some damage modifiers mentioned in the Forge are not included above. This is because they are extra bonuses which apply in addition to the damage types in the above slots.
These extra bonuses are Ultimate, Melee, Combo, Blast, and Item Level Scaling.