The Importance of Armor
This Summer, the metagame was quite unusual. With the rise of the Disco Pony, aka 
Leshrac, we saw a metagame centered around magic and pure damage, despite 
Glimmer Cape being introduced this year. 
Storm Spirit, 
Lina and 
Queen of Pain were quite dominant throughout that phase.
Now however, we have a new patch and a new meta on our hands. 
Alchemist has gained an increasing amount of popularity, as the meta swings back to a more physical DPS based one. Guest-writer GorgonTheWondercow talks us through Armor, one key mechanic in the game when it comes to understanding physical damage.
What is Armor?
Armor takes all incoming physical damage and cuts out a percentage of it. There are some generally unimportant caveats regarding damage types. On the surface, armor looks difficult to understand. It really isn't.
Think of armor as increasing your own effective hitpoints instead of reducing incoming damage. For every one armor, a hero gains 6% of his current HP in against physical damage. So if you have 100 health and 1 armor, you can take 106 physical damage without dying. With 100 health and 20 armor, you can take 220 damage before dying.
I will use "effective hp", "effective health" or short "EHP" in reference to the amount of damage a hero can take from physical damage.
The Armor Era
As the meta swings back toward DPS-based cores ( Alchemist, 
Windranger, 
Spectre, 
Legion Commander are at the upper echelon of pickrate growth in pub Dota), armor and negative armor rise with them.
Applying negative armor increases your team’s overall physical damage output, so heroes such as 
Templar Assassin, Alchemist, and 
Slardar are on the rise alongside physical-damage drafts. For heroes like 
Dazzle or 
Lich, drafting positive armor when your opponents are likely to deal massive physical damage is central to survival.
Dota is an ecosystem and no hero’s popularity is truly one-dimensional, but armor's role in the current metagame has been a massive contributor to trending heroes.
Alliance vs Ninjas in Pyjamas game two from the Frankfurt qualifiers had a memorable example of negative armor's vital role (warning: spoilers in this section).
At 49:50, Alliance barely clings to life with their ancient under assault. Akke drops a death ward, which has a physical damage output, as NiP's Era tries to attack the ancient down. Alongside an army of creeps it will take him about 8 – 10 seconds to kill the ancient. Death ward would normally take about nine second to kill him given his EHP; the ancient should fall with a narrow margin of success. But Loda had already dropped Acid Spray, which helped zone NiP's heroes who didn’t want to stand in the Death Ward/Acid Spray combo. Now Era takes about seven seconds to die (without factoring regen).
In fact, none of NiP's heroes would live long enough to kill the ancient if they walked into the combination. The Acid Spray's negative armor helped create a situation where a comeback was possible.
Advanced Armor
Armor increases the value of raw HP and vice-versa. Each armor will always raise your EHP by 6% of your current HP, but that doesn't scale linearly when you consider EHP growth. There are diminishing returns to EHP growth from purchasing armor.
Warning: math to prove the point; you can skip it if you already believe me.
Let's imagine a hero with 1000 health and 0 armor. His effective HP is 1000. Now he buys 5 armor. His effective HP grows by 30% of his base HP, which is 300. He has 1300 EHP. He buys 5 more armor. His effective HP grows by 30% of his base HP again, still 300. He has 1600 EHP. In terms of EHP value, his health is going up linearly. In terms of his percentage growth of EHP, he is gaining less with each additional armor. The first five armor gave him 30% (300/1000) but the second five armor gave him 23% EHP growth (300/1300).
When Should I Buy Armor?
Casual Armor
It's okay to just buy an inexpensive, casual piece of armor for survivability and sell it later, especially for tanky heroes with an extra item slot. That option is far better than dying because you greedily saved to get a big-ticket purchase thirty seconds faster.
The Ratio Rule
Armor is typically better than HP if you have more than 1400 health. The exception is if you're against a heavily non-physical damage opponent, such as 
Silencer or [missing hero: outworld-devourer]. Armor has the best price-to-effect ratio in the mid game, but the highest impact in the late game. On average, armor items give about 1.6% effective hitpoints per 100 gold spent. Later game items do slightly better. HP and strength-based items give about 13 health per 100 gold spent in the early game, about half that in the mid game, and about double that in the late game.
The exact measurements will differ per item, this is just a loose guildeline. This is just to account for survivability, but doesn’t take into account other benefits of items.
So if you cross 800 base HP, it will be more cost effective to build items such as 
Ring of Basilius than 
Bracer to deal with physical damage. If you have 625 HP, it will be more cost-effective to buy items like 
Mekansm than 
Drum of Endurance. If you have 1400 HP, it will be more cost-effective to build late-game armor such as 
Shiva's Guard than late-game HP such as 
Eye of Skadi or 
Heart of Tarrasque. While in lane, armor is best if you have at least 775 HP. For pushing and teamfighting, it’s best if you have at least 435 HP.
The Nuclear Option
When deciding if you should buy armor, ask yourself how much health you’ll have after your opponents use their magic damage. Don’t factor your armor decisions on your max HP bar because you'll rarely fight without taking any magical or pure damage. Overinvesting in armor leaves you vulnerable to magical or pure-damage nukes.
For example, you're playing against a 
Lion with a level one Finger of Death. Add 625 HP to the HP you’d lose against Finger of Death (425) and 
Chainmail will typically give more survivability than Bracer at 1050 HP.
Jungle Sustain
Heroes who want to go to the jungle early benefit tremendously from small boosts of armor. Even a cheap 
Ring of Protection can extend your jungle sustain and will maximize the value of any health regeneration items. Since armor increases EHP as a percentage of your base HP, it also means that armor increases the value of consumables.
With no armor, the 400 HP from a salve offsets exactly 400 damage. With 5 armor, the 400 HP from a salve offsets 520 damage so it provides 30% more sustain while tanking creeps. Heroes who will tank jungle camps from level three or four and rely on the use of consumable regeneration, armor is an absolute must.
Armor on Strength Heroes
Strength heroes' high health pools make them perfect armor bearers. 
Tiny and 
Treant Protector are all examples of heroes whose core attributes and typical item builds give boatloads of raw hitpoints, and those heroes benefit the most from armor boosts.
A Ring of Protection on a level 12 Tiny with 
Aghanim's Scepter and Strength 
Power Treads will give him 330 more hit points against physical attacks. That’s seven times more cost effective than a 
Vitality Booster.
Armor on Agility Heroes
Heroes with high agility growth, such as 
Anti-Mage, 
Weaver, or 
Phantom Lancer will typically want to stay away from purchasing armor directly. These heroes tend to have very low strength gain, meaning they have low base HP.
In order to maximize their already great armor, they need to increase their HP (items such as Eye of Skadi, Heart of Tarrasque or 
Octarine Core). This is especially true later in the game where agility heroes will naturally have 16 – 25 armor. Agility carries also often build 
Manta Style. Although illusions retain the hero's base armor—so they benefit from agility increases—they do not benefit from armor purchases. This makes HP gains far more effective at increasing illusions' survivability.
Every seven points of agility give an agility hero one armor, in addition to increasing their attack speed and base damage, so if you need more armor it's typically more cost effective to purchase agility instead.
Armor on Intelligence Heroes
Int heroes tend to have moderate strength gain and low agility gain. They also typically avoid direct line of fire, trying to work around edges of fights, or initiate then fade back. Support Intelligence heroes should buy forms of armor that also help teammates, such as Ring of Basilius or 
Medallion of Courage. Consider this: a Ring of Basilius adds 12% hitpoints to all nearby allies for 500 gold. If your team has a total HP of 4000, the Basilius Aura will add 480 effective hitpoints. If you have more money to spare, you should of course never forget about 
Ghost Scepter, the ultimate protection against physical damage.
Cores usually benefit more from prioritizing raw HP or HP regen in order to be able to sustain damage (Octarine Core Leshrac, 
Bloodstone Storm Spirit, or Aghanim's Scepter Windranger).
Dealing with Armor
How to we stop tanky opponents who make strong build choices? Part of this comes down to draft and part to item builds, but here are the basic tools for those decisions. Heroes with both high health and high armor should have their armor reduced. Generally save nuke damage for more susceptible targets.
Heroes with high health and low armor can be handled by increasing physical damage alone.
Opponents with low health and high armor are the best targets for magical or pure damage nukes. Locking these heroes down is vital to killing them with attacks, but often not necessary if one or two solid bursts of magical damage are available (especially before Spell Shield or other forms of protection are available).
If your opponent has low health and low armor, you can kill him any way you want.
Is 
Solar Crest/ Medallion of Courage Still Good?
Yes, especially on Dire. Behind Ring of Basilius, Glimmer Cape, and Mekansm, Medallion of Courage is the most purchased support item in pro Dota. Medallion of Courage only has a 42.5% winrate in pubs, but that is because the losing team playing from behind is often the team that can’t afford to upgrade it to Solar Crest.
When you compare Solar Crest to similarly priced items (
Rod of Atos, 
Crimson Guard, or 
Pipe of Insight) it comes out ahead in success rate at about 65%. On the Dire side, a Medallion of Courage/Solar Crest can turn almost any successful fight into a free Aegis because of Dire’s ease of access to the pit.
As a caster, I’ve seen a lot of games lost because a Dire didn't have the negative armor to finish Roshan quickly enough. Why are these items so key as physical damage dealers become more prominent?
Against a 1500 health hero, Medallion of Courage reduces effective hitpoints by 630. In cases where units are guaranteed to stand still (for example, if they are Dueled), these items effectively act as after-the-fact nukes, like Orchid Malevolence. If you cast Medallion of Courage onto a hero who dies to Duel, you effectively hit him with a nuke worth 42% of his hitpoints.
Postscript (Mathing Armor with proof)
Formulas:
Damage multiplier = 1 – (0.06 * armor) ÷ (1 + 0.06 * |armor|)
Effective HP = Total HP * (1 + Armor points * 0.06)
Version of Alliance example from metagame section with math included: At 49:50, Alliance is clinging to life with their ancient under assault. Akke drops a death ward, which has a physical damage output, as NiP’s Era tries to attack the ancient down. With 2566 health and 24 armor, he would normally have 6261 effective HP against physical damage.
He’s dealing over 300 damage per second, so alongside an army of creeps it will take him about 8 – 10 seconds to kill the ancient, less with teammates. Death ward does 682 damage per second to each hero it hits. That means it would take about nine second to kill him. A very narrow margin of success on that defense. But he’s standing in Acid Spray, courtesy of Alliance’s Loda. That cuts his EHP to 5183 and allows them to kill him in about seven seconds. Better yet, any hero from NiP within attack range of the ancient is forced to stand in that acid, reducing all of their effective HP enough that the bounces from death ward would kill any of them before the ancient falls.
You also have to take in account that armor not only raises effective HP, but also things like effective heal and effective regeneration (important on alch). Armor works really well when you have some healers on your team.
p.s.: same goes for life steal.
@lForeveRl If you use aghs on dazzle and another armor reduction hero (alch,weaver,tidehunter,naga,TA,venge,CK,ET,invoker's spirits) you can easily reach that -100 armor.