In D&D 5th Edition, your "Armor Class" is a sort of generalized amalgamation of how hard it is to hit you. The reasons why it might be difficult vary - you could imagine that a dextrous rogue is just dodging out of the way of things, while a monk uses their dexterity in combination with a heightened awareness of their surroundings to dodge more effectively (hence why they add their wisdom to their armor class). A heavily-armored fighter isn't so much not getting hit by things as much as they are blocking blows with their shield or a monster's claws are glancing off their tough armor plates.
While some long-range casters might not care much about their AC, generally any character is going to want to try to build their character around ensuring that their armor is maximized.
Now, having come to D&D from World of Warcraft, there's a bit of a parallel - in WoW, the four types of armor are Cloth, Leather, Mail, and Plate, which corresponds decently with 5E's No armor, Light Armor, Medium Armor, and Heavy Armor.
There are only three classes in 5E that are not proficient in any armor: the Wizard, Sorcerer, and Monk. Barbarians are actually proficient in Light and Medium, but they, like the Monk, get a special "unarmored defense" feature that allows them to get their AC up to decent levels despite not wearing any protective gear, so you could count them in here.
A typical character's AC is calculated by adding 10 plus their Dexterity modifier. Because you have no protective gear, the only way you're avoiding getting hit is dodging out of the way. Wizards and Sorcerers can take the Mage Armor spell, which for a single 1st level spell slot bumps that 10 up to a 13, which actually makes for a better armor class than even the best non-magical Light Armor, lasting up to 8 hours with no concentration.
Monks get to add their Wisdom modifier on top of that, while Barbarians can add their Constitution modifier if they're not wearing any armor other than a shield. I like to think that while the Monk is using a powerfully honed awareness to know exactly when to dodge incoming attacks, the Barbarian is just so tough that weapons or spells kind of bounce off their insanely tough bodies sometimes.
For Rogues, Bards, most Warlocks, (and Druids, who start with just light armor even though they can use Medium, as long as it's not made of metal) the options expand to Light Armor. While there is cheap Padded Armor, you'll almost never start with that given that these classes start with Leather Armor. While wearing Leather, you calculate your armor as 11 + Dexterity, and you can upgrade to Studded Leather Armor to change that 11 to a 12. This armor works well for dexterous characters, as you'll be able to get up to 17 AC once you've maxed out at 20 dexterity (or 19 with a shield.)
Light armor, except for the "why would anyone use this" Padded variety, imposes no disadvantage on Stealth checks, which is good given that it's the armor Rogues tend to use. Still, it's not much of a boost compared to heavier forms of armor, and still relies on having high dexterity, which might not be as useful for Bards or Warlocks who want to focus on their Charisma.
Now we come to Medium Armor. This is the armor that about half of the Clerics, Druids who find some made of organic material, Rangers, and Artificers use. Medium armor is also the most mechanically tricky (not that it's very tricky) type.
Like Light armor, there's a low-grade version called Hide that you're probably not going to bother with. Instead, the three typical examples are Scale Mail, Breastplates, and Half-Plate. With Medium Armor, you start with a base number and add your Dexterity modifier, but the cap on the bonus that your Dexterity adds is 2. So if you have 16 Dexterity, you would only add 2 to the base, rather than 3.
What's nice about this is that you don't need very high dexterity, because the base number is pretty high. Scale Mail and Breastplates start at 14, allowing you to get up to 16 based on your Dexterity. Half-Plate starts at 15, for a max of 17. The downside here is that Scale Mail and Half-Plate imposes disadvantage on Dexterity-based Stealth checks (which is most Stealth checks,) so it's hard to be sneaky wearing this kind of armor. Still, if you don't care much about being sneaky, there's really no downside. Even if you're a pure Dex character (like a Ranger, say,) you'll still be getting just as much Armor Class with Half-Plate as you would with Studded Leather, and if you can get that before you've maxed out your Dexterity, you'll get it earlier.
There is a feat you can take, Medium Armor Master, which does some nice things: it raises the Dexterity contribution here to 3 and also prevents the armor from interfering with your Stealth. The upshot of this you actually wind up being just as well armored as someone in the heaviest of heavy armor, but with no stealth penalty - but it only works if you've got the Dexterity to burn (again, if you're a Dex-based ranger, or the Artificer I made that rolled really good stats, you'll be golden.)
Finally, we come to heavy armor. Fighters, Paladins, and about half of Clerics get this to start with. Is Dexterity your dump stat (my Paladin has a 7)? This is the armor for you, because it's entirely divorced from your Dexterity modifier, and is just a flat number. Heavy armor comes in four varieties: Chain Shirts (which are, like Padded and Hide, the cheapo stuff you'll typically only find on NPCs) Chain Mail, Splint, and Plate. And these each grant 15, 16, 17, and 18 armor, respectively. What's the downside?
Well, first off, all of them impose disadvantage on stealth checks, as you'll be clanking around in heavy metal. The other is that they do require a prerequisite amount of Strength. Chain Mail requires 13 Strength while Splint and Plate require 15. If you don't meet the requirement, your movement speed is reduced by 10 feet if you try to wear it.
For Fighters and Paladins, this is unlikely to be an issue, as you'll be pushing Strength as your primary stat (there are some Dex-based fighters, and even a few Paladins who go that way, and those who do might consider going with Medium or even Light armor instead.) Clerics, however, will need to consider if they want to commit that much to Strength. That being said, most Cleric domains that use heavy armor also get a bonus to their melee attacks (typically adding some kind of magic damage) at level 8, so if you'd like to whack things with a mace instead of only casting spells, you can pump Strength as your secondary ability after Wisdom and not have to worry about this potential downside.
The other major downside is that heavy armor is expensive. For top-of-the line armor, you've got Studded Leather, which costs just 45 gold, Half-Plate is a big jump to 750 gold, and Plate is 1,500 gold, which, unless you've got a very generous DM, means you probably won't be hitting that amount of gold until you're well into tier 2 at least.
Still, the benefit of heavy armor is that, if you get Plate, you actually get the highest possible AC that armor provides. Unless you find magical ways to boost a character's Dexterity beyond 20, the best you can get with (mundane) Studded Leather armor or Half Plate is 17, while Plate grants 18. Yes, a Monk or Barbarian who maxes Dexterity and Wisdom/Constitution (respectively) will get 20 AC, but unless they rolled god-stats, they are very unlikely to be able to do that until at very high levels.
So, breaking this down: what sort of armor should you take, and how should you build your character around that?
The first question to ask is whether you really care much about AC. If you're a caster who always stays at the back, you will sometimes have to worry about a melee fighter breaking through the lines or sneaking up on you, and there are ranged spell attacks, but if you're good at staying away form the fray, you will probably be ok not worrying too much. On the squishiest of casters I typically just favor Constitution over Dexterity to make sure that whatever hits me, I'm likely to survive it.
Most classes don't really have a choice. I do recommend, however, that if you can choose between Light or Medium armor, you should go with the latter unless you really need to be stealthy (like if you're a Rogue.) Even if you're a Dex-based character, Studded Leather only starts to outperform Scale Mail once you've maxed out your Dexterity.
If you want a sword-and-board melee fighter, you can go with Dex and take a Rapier, which is just as powerful as any other weapon wielded in one hand, along with having finesse. But I do recommend that if you want to max out your tankiness, you should probably go strength-based and take heavy armor to squeeze every drop of AC out of your gear. You'll also, naturally, want to do this if you're going for any of the heavy, two-handed weapons like a Greatsword or Maul.
Ultimately, the margins here aren't massive - one more AC just means the DM needs to roll one number higher on the die, and when you start fighting monsters with a +12 to hit or so, even having your AC way up at 21 (like my Fighter currently has,) they're still going to be hit sometimes (though he's an Eldritch Knight who has the spell Shield, so I make it very hard for the DM to hit me.) But in the long run, your AC will make you last longer in fights, and especially given that when those monsters with massive to hit bonuses start hitting you with attacks that do way more than typical weapon damage, you'll be very grateful when the attack does miss.
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