Borderlands – How does “Bonus Elemental Damage” work

It's an additional splash damage amount in addition to the weapon card damage. Percentages vary by weapon type.

Some Maliwan weapons list "Deals Bonus Elemental Damage" as an effect. These weapons deal an additional portion of elemental damage as splash damage around the bullet impact. The splash damage equals 50% of the weapon card damage for sniper rifles and Plasma Casters, and 80% for pistols; it is multiplied as normal for the element type, but is not further multiplied by critical hits or badass rank "elemental effect damage." Amp shield damage only applies on direct hits, and not this splash damage. However, the splash damage can trigger an elemental damage over time effect independently of the main bullet, resulting in double DoT triggers on some shots. This effect is especially useful on slag-snipers as the elemental explosion is guaranteed to slag the target even on a non-direct hit, as opposed to non-Maliwan sniper rifles that need to hit the target in addition to having only a reduced chance of applying the slag effect.

Rocket Launchers with "Deals Bonus Elemental Damage" are a special case. Whereas normal rocket launchers simply deal their weapon card damage as splash damage in a certain radius, DBED rocket launchers have two separate splash ranges. The smaller splash range deals 50% of the weapon card damage, and the larger splash range deals 25% of the weapon card damage. These stack, however, so anything within the smaller range is also hit by the larger range, resulting in a total of 75% weapon card damage on close hits. As with pistols and sniper rifles, these two damage effects can trigger the damage over time effect independently.

^{Source: Borderlands Wiki, "Borderlands 2 Elemental Damage - Bonus Elemental Damage"}