Explosive damage is one of the four main damage types encountered in space combat, and usually shown with Yellow icons and missile tips.
In general, Shields have very strong resistance to Explosive weaponary (typically ~60% resist), but ship Armor does not do well against it (typically ~10% resist). These resistances may be increased with the appropriate active or passive Hardener modules.
Angel Cartel NPC Pirates commonly use projectile weapons, and so can be expected to deal Explosive damage.
Tactically, Explosive weapons are really only useful in a 'deathblow' capacity, as any shielded craft will strongly resist the incoming damage. Best used in concert with Electromagnetic weapons, they can however, do an impressive job on armor, and can make a significant difference against any ships suspected of Armor Tanking.