@ Noir_CZ
Piercing - Standard Armor Piercing from vanilla. Basic ballistic rounds you would find in conventional weapons. Rifles have a slight bit of armor negation with this, and because this is a common early form of damage, many higher tier enemies have in built resistances to it (I'm looking at you, Mercs).
Slash - Melee, also known as Cutting. Almost always on melee or thrown weapons: Axes, Swords, Throwing Knives etc. Usually very solid damage in its Melee form, and often scales up in thrown form with throwing skill (more skilled archers get more damage, etc). Usually a solid early damage type as most enemies don't have extra resistances to it. (This is actually one type Mercs are more responsive to).
Concusive - High Explosive from Vanilla. Great for terrain damage. Often useful as bombs that use it hit the under armor of the targetted unit. Does carry some secondary stun effects, but in practice is more likely to kill targets if it overcomes their armor than stun them. Higher tier enemies are often resistant to this, but as with anything, apply it enough, and you may still get effects.
Daze - Stun damage. Knocks enemies unconcious. With OpenXcom Extended, the waters become a bit more muddied here, as weapons can have a primary damage type and secondary damage effects. For example, the Handle is primary Daze/Stun with secondary lethal effects. This means that as damage scales up with your strong gals, you may sometimes kill a target you were trying to stun with a handle. Also, some targets are immune to Daze/Stun damage. Another mechanic in OXCE allows other weapons that are designed for incapacitating enemies. These are often listed as "Plasma" damage in the Bootypedia, but they have extremely minor lethal damage and 4x stun damage. These are often later in the game.
Choking - Smoke from Vanilla. Please note that almost all units that don't have environmental filters (masked helmets, etc) take 4X damage from smoke as compared to vanilla. You can see this in your own gal's base stats. Does stun damage over time. Now, throw in OXCE mixes on assorted weapons as mentioned above in Daze, and you can get Smoke type weapons that also do significant Lethal damage (to health) or pain (often to Morale) or even to energy and time units. Most early Choking stuff is just smoke though.
Chem - Same as Acid from vanilla (ie Celatid spit). In XPirateZ almost always degrades armor, usually adds pain (Morale damage) and has good Terrain Destruction effects.
Plasma - Plasma from vanilla. " sometimes plasma damage is stun (cattle prod, tazogun?)" - See explanation in Daze above. Most enemies will respond to doses of plasma as very few are resistant to it. However, access to good plasma weapons for the player can be fairly limited for sometime.
Laser - Laser from Vanilla. This always has some armor penetration properties so it can be useful against armored enemies. Again, not easy to get a hold of early on.
Not mentioned:
Burn - Incendiary damage from vanilla. If a target is hit Fire will almost always damage the target. If the target is set on fire from being hit or starting its turn on a tile on fire, it will continue to take damage from the flames. Some units are immune, while others are very vulnerable (Academy Drones). Often targets under armor so this too helps with the damage it applies.
Special - This is listed and is a rare damage type applied via Voodoo(Psi) weapons. Has varied effects ie, stun, morale, tu, energy, armor damage, in addition to possible lethal damage.
Cold - A weather affect of environmental conditions, causes energy drain on player units not wearing little or nothing. (Bikinis are bad in the Arctic)
Heat - Another weather affect: Caused stun on units that might overheat in Jungles or Desert.
Acid Rain - Damages armor, and morale. May cause stun?