Damage types done by fations.

Angel: All types, although mostly kinetic and explosive.
Amarr Navy: EM and thermal.
Ammatar Navy: EM and thermal.
Blood: EM and thermal.
Caldari Navy: Kinetic and thermal.
Centii: EM and thermal.
Coreli: Kinetic and thermal.
Corpii: EM and thermal.
Domination: All types, mostly kinetic and explosive.
Gallente Navy: EM and thermal.
Gist: All types, mostly kinetic and explosive.
Gurista: Kinetic and thermal.
Khanid: EM and thermal.
Mercenary: All types.
Mordus: Kinetic and thermal.
Odamian: Kinetic and thermal.
Pith: Kinetic and thermal.
Rogue drones: All types.
Republic fleet: All types, mostly kinetic and explosive.
Sansha: EM and thermal.
Serpentis: Kinetic and thermal.

Main Damage types to use on npc factions:
Guristas - Kinetic
Serpentis - Thermal
Mordus - Thermal & Kinetic
Blood - EM & Thermal
Sansha - EM & Thermal
Angel - Explosive
Mercenary - Thermal
Corpum - EM & Thermal
Gist - Explosive