Some weapons have a skill called [fancy adjective] assault where the adjective is dependent on the element and power of the skill. Those weapons give what the community generally refer to as assault values. They count towards any himes whose element matches that of the weapon.
The assault value is X+0.5*weapon skill level (NOT to be confused with the weapon level - the skill level number appears next to the skill when you check its details and goes up to 20, except for FLB weapons, which go up to 30.) X depends on the power of the assault skill - small(no plus sign next to skill) = 0%, medium (+) = 3%, large (++) = 6%.
Unfortunately, the game does not sum up the assault values for you, so you have to go and check every equipped weapon and do the math yourself to come up with it. Note that there is a closely related skill called pride that also gives assault values, but is dependent on hp - lower the hp higher the assault.
A highly simplified dmg formula is dmg = base atk * (1 + assault) * (1 + elemental) * (a bunch of other stuff that's probably beyond the scope of the current discussion.)
That formula is the basis for Slashley's comment about you wanting to boost what you have less of, all else being equal (the disagreement is really about when all else is not equal and which situations actually matter if I am not misreading everyone's comments.)