Physical Damage
We have 3 physical damage types that (almost) all weapons have, in different amounts. Here they are:
Impact - Most effective on shields
Puncture - Most effective on armor
Slash - Most effective on health/flesh (non-robotic)
That seems simple enough, right? Well, those of you who have been enjoying Pokemon Snap in space, if you check the enemies in your Wardex (Codex), you may notice:
Corpus - Have health and shields, but NO armor
Grineer - Have health and armor, but NO shields
Infested - Have health, but NO armor OR shields
Using those 2 key pieces of knowledge about damage types and enemy types, you should now be able to look at your weapons properly. Let's take a few examples, starting with the loudest complaint: Soma
Using what you just learned about damage types, you SHOULD be able to see that this is probably never going to be a good weapon to use against high level Corpus. Why? Because Corpus rely on heavy shields which are weak to impact damage, of which Soma sadly has little to begin with. Soma is fairly even on puncture and slash however, with high crit. This means you should boost your puncture/slash damage, slap on some crit mods, and you'll be able to destroy Grineer and Infested. Soma is now most efficient and effective against Grineer and Infested, so stop taking it into missions against Corpus and expecting miracles.
So now looking at the Boar, out of the 3 physical damages, it has high impact and medium slash damage. Judging from that alone, your most beneficial build will be as an anti-Corpus weapon, because Corpus shields are weak against impact damage, and Crewmen's health is weak to Slash. Also, the extra slash damage Boar has is nice, because when elemental damages are calculated, they're based on the percentage of your total physical damage (Impact + Puncture + Slash). So it may be worthwhile to boost Boar's slash damage as well.
Despair has a large chunk of base puncture damage, which is most effective on armor. The proper build here would be focusing on boosting puncture, slapping on a max No Return mod for +60% puncture damage. Why not use a max Razor Shot mod too, so I can boost Slash damage by 60%? Well, look at the math:
Base puncture damage * 60% is equal to 44 * 1.6 = 70.4 damage
Base slash damage * 60% is equal to 8.2 * 1.6 = 13.12 damage... kinda feels like you wasted power points by using this mod for an extra ~5 damage, huh?
Here's a quick cheat sheet for determining builds at a glance (Updated for 11.0.7):
Elemental Damage
There are 10 possible elemental types, but that doesn't mean you should use all of them at once. Just like with the physical damage types, some elemental types are more useful on certain weapons than others. Here's a quick overview of what they are and how they combine:
Well, if you played with the old damage system, you know Corpus shields are weak against Cold, and they can be stunned by Electricity if they don't have shields up. Infested have resistance to electricity, but are particularly weak to Heat. Grineer are really tough, but Corrosive causes heavy damage to their thick armor, leaving their exposed flesh open to Heat, Viral, and Blast damage.
The number one rule with Damage 2.0 is:
Use the right tool for the job, and compliment your weapon's base stats.
Learn to think of your weapons as anti-Corpus, anti-Grineer, or anti-Infested. When you start viewing it like that, it becomes completely obvious that you need to use an elemental that Corpus are weak against on your anti-Corpus weapon. If you've got a high puncture weapon, you're going to want the Corrosive element as well, because that's what Grineer are weak against. A heavy impact weapon? That's great for Corpus, so add on some Magnetic element. Slash weapons are particularly useful for... you guessed it, Infested! Simply add some Heat and Blast damage and go eradicate some Infested areas of the galaxy.
The primary elementals are fairly straightforward, but I also like to divide the combined elementals into 2 categories:
Utility
Blast - Stun/knockdown
Gas - Poison AoE
Radiation - Confusion and slowed fire rate/reduced accuracy
Direct
Magnetic - Shield reduction
Viral - Health reduction
Corrosive - Armor reduction/removal
Utility is most useful as a secondary effect, on top of a primary effect you've given your weapon. For example, Embolist already defaults to Toxin damage (a primary), and it's really nice to have Blast (secondary from fire and ice) on top of that to keep things locked down.
Spoiler
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Hopefully that explains most of what DE Scott intended, and it really is an elegant design for weapons and enemies, once the kinks are worked out and it's explained to players better in-game. Any more questions?
Updates:
DE_Steve has suggested Radiation against Grineer, and testing it out with Synapse shows little penetration through Armor, but to non-armored Grineer health it's fairly effective. (Although in-game codex states no weakness)
Added FAQ below based on questions in this thread.
Updated information further based on testing and feedback (Thanks to: Phoenix86, Liacu, jrkong, NegimaSonic, MeduSalem)
FAQ Follow-up Questions
Spoiler
Q: Is it true that there are no more armor-ignoring damage types in Damage 2.0?
A: Boost your weapon's Puncture to gain better damage against armored enemies. You can also use the Corrosive element to reduce/remove armor, along with the Aura mod, Corrosive Projection.
Q: Do bullet type damage not work on Pure Elemental weapons (Ignis, Synpase, etc.) or was that a UI thing?
A: The unique elemental weapons (Ogris, Ignis, Synapse, etc.) use their elemental damage type as default, instead of physical damage. You can still increase this damage by using same element and generic damage mods like Serration. Example: Ogris is 300 Blast damage base, equipped with max Serration (+165%) gives it 300 * 2.65 = 795 Blast damage.
Q: Also how does status chance work? Does it allow for every damage type that the weapon has to hit special effects?
A: In patch 11.0.2, they added icons to damage numbers indicating when Statuses proc from your weapons. You'll notice the status chance is somewhat low for certain weapons, and higher for others. Unfortunately now, just because you're using Cold on your Kunai, it doesn't mean you'll be freezing enemies with every shot; the Frozen/Slowed status effect only has a small chance to proc, depending on the weapon's Status Chance. A further note, is that most of the time when a Physical status procs, it will be the status of type of damage that you have the most of on that weapon. So if you have high impact on a weapon, if a physical status procs, it will most likely be Impact's stun/knockback.
Q: What does armor/puncture damage amount do, exactly? Do enemies now have a set amount of armor (like shields and health) that you blast through before doing damage to health? Or does armor prevent X% of damage from reaching health, and when you deplete the armor you do 100% health damage?
A: Armor does not deplete like shields, but you can reduce/remove an enemy's armor using Corrosive status effects, the Corrosive Projection aura mod, and possibly other mods/abilities. Armor works like mitigation and prevents X % of damage from reaching health, so you'd probably want more puncture than the other two physical types.
Q: Is slash the only physical damage type that lowers health, essentially making the other physical damages useless on low level Infested?
A: No, all damage types are supposed to be able to damage health, but Slash is more effective at reducing non-robotic health fast. It's also cautioned that Impact and Puncture are up to 50% less effective against Infested, meaning you would be better using a weapon whose physical damage was mostly focused on Slash.
Q: So we're pretty much forced to go a one build kind of way with weapons?
A: Somewhat, but not really. For physical damage you're usually better off complimenting your weapon's base stats, but you can find a lot of versatility in the effects you put on it. Technically you could also fit up to 3 elemental effects on your weapon, utility and otherwise, but they may be competing for proc chance (we're not sure yet). It's really just up to your personal preference at the time, what you want to go for...
For example, Synapse is a beautifully versatile weapon. It defaults as electric damage, which can be:
Boosted purely as Electric, or
Combined with Cold to form Magnetic and drain Corpus shields fast, or
Combined with Heat to form Radiation to debilitate and confuse most enemies, or
Combined with Toxin to form Corrosive and melt Grineer armor, or
Combined with one of the above and Blast/Gas for even more crowd control possibilities...