In Dungeons & Dragons 5th edition, what is the best race for a character that is a barbarian?
"Best" would be defined as the following:
I'll give a few of the best races with pros and cons.
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(Legend: PHB -> Player's Handbook, EEPC -> Elemental Evil Player's Companion, VGM -> Volo's Guide to Monsters)
There are other races that can work, but in terms of an optimal meshing of features and flavour, these four are the best of the best.
A tribe of rampaging Forest Gnome dex-based Bear Totem Barbarians would take over the world. They're resistant to all damage when it matters (except psychic). At 2nd level, they have either advantage or proficiency in pretty much any saving throw they will ever need to make. A Gnome Bear-barian will always be the last man standing.
They've got the nature connection. Any of them that aren't raging at any given moment can add special effects to freak out opponents, via Minor Illusion at will.
Max out Dex and Con, don't worry much about Str or the +2 raging damage they won't use (3rd choice for stat placement). They'll have good AC, lots of hit points, and Reckless Attack if they need to hit a difficult target. Fast movement will help tactically. Rapier and shield, or dual-wielding handaxes or daggers -- unstoppable.
Have a few change class to Wizard, to buff the tribe and provide artillery, and you've got perfection.
Looking at regular PH races exclusively, Half-Orcs are, from my experience, the most "suited" race. They often live in tribe-like communities, which fits to the way barbarians work in 5e roleplay-wise.
Also, stat-wise, they have initial strength and con bonuses - the most important stats for a barbarian -, they have "Relentless Endurance" (basically, you don't die when dropping to 0 hit points, 1/day), and they have savage attacks, so you deal extra damage on a crit.
Both of the latter abilities are also available to barbarians at higher levels - except they stack with a half-orcs abilities, so your barbarian gets even better at what he does.
Furthermore, proficiency in "intimidation" is pretty nice to have as a barbarian.
Other decent races include dwarves - you gain poison resistance and advantage on saving throws against poison, which increases your survivability, and you also increase your con and potentially strength stats.
Also, there's a dwarf-exclusive barbarian subclass, which could be interesting. I personally prefer the "standard" berserker, but to each their own.
Dragonborn also get a Strength increase, they get resistance to their respective dragon type, and they have a breath weapon, which can be potentially pretty useful (especially cone effects) since you're probably in melee anyway.
Aasimar get resistance to necrotic and radiant damage, and the 1/day flight ability of the protector subrace or the 1/day spheric radiant damage around the player of the Scourge subrace can be pretty nice as well. Note that the stat bonuses of this race are not optimal - the base race gives you +2 charisma, which is (for you) mainly useful for intimidation, which you might often be allowed to do as a strength-based check. Protectors get +1 WIS, Scourge Aasimar get +1 CON (which is useful), and Fallen Aasimar get +1 STR, obviously also helpful.
Hence, overall, they are only a mediocre race for barbarians.
I will add details for other non-PH races later (i.e. Volo's Guide, where Aasimar are from), since I currently don't have any rulebooks on me and recalled this information from memory.
I'm guessing the "Orc" playable race from Volo's Guide will be pretty nice as well, but I haven't really taken a look at that yet, so I don't know without a rulebook in front of me.
There we go:
Earth Genasi (Elemental Evil): +1 STR, +2 CON; they don't have any actual abilities that are of great use to barbarians, so not really the best race.
Goliath (EE or Volo): Easily one of the best races for barbarians aside from Half-Orcs; they get +2 STR and +1 CON, gain Athletics proficiency, can shrug off 1d12+CON damage once per short or long rest, and a few minor other abilities.
Bugbears (Volo): +2 STR and +1 DEX, +5ft weapon range on attacks during your turn, Darkvision, stealthy and their own version of sneak attack once/combat.
Orcs (Volo): Surprisingly less efficient than Half-Orcs. They gain +2 STR and +1 CON, can move up to their speed towards an enemy as a bonus action, gain Intimidation proficiency and darkvision.
If you're going to go for the Totem Warrior subclass and choose the bear totem at level 3, ignore anything mentioning resistance in this answer. Since you will get resistance to all damage types except for psychic, racial resistances get pretty pointless. The cases where you will take damage while outside of a rage are pretty rare, and it's not very likely that the few resistances you get from your race will be relevant in that case.
This answer presumes that your DM permits the Variant Human (Adventure League allows that at character creation).
Racial +2 Str, +1 Con
Racial added Proficiency in Intimidation
Racial Relentless Endurance (avoid 0 HP once per long rest, which means staying in the fight)
Racial Savage Attack synergizes with class Brutal Critical feature such that for DPR focused builds this is hard to beat.
Racial Darkvision
As noted in other answers, the Half Orc is a near perfect fit for Barbarian.
A build using Point Buy: 15 15 15 8 8 8
Translates to: STR 16 DEX 16(Feat) CON 16 INT 8 WIS 8 CHA 8
A key advantage to this build is that you get proficiency to Dexterity saves, which is a very common save. (For example, versus dragon breath or a fireball). This adds up to proficiency in Dex, Con and Str saves: a very melee/combat friendly save package that will boost your scores as you go up in level and face increasingly more dangerous monsters.
Danger Sense
At 2nd level, you gain an uncanny sense of when things nearby aren’t as they should be, giving you an edge when you dodge away from danger. You have advantage on Dexterity saving throws against effects that you can see, such as traps and spells. To gain this benefit, you can’t be blinded, deafened, or incapacitated.
Adding proficiency to your advantage and you'll likely not miss many Dexterity saves versus traps and spells.
Your unarmored Defense without a shield is 16 and with a shield is 18.
If you boost Constitution with an ASI (which for tanky characters is common) your AC also goes up. Synergy.
Rage works very well in this build: a tanky front liner whose Dexterity saves versus spells and traps goes up with level.
A decent list of skills is Athletics, Perception (which will offset low wisdom) Intimidation and Survival but that will depend on your setting and your own skill choices.
Drawbacks:
No darkvision, weak wisdom saves (a common save versus magical effects), one less proficiency than the half orc, and reduced Intimidation chance as a class skill if you choose that proficiency.
If your DM goes along with the PHB based idea of using Strength on some Intimidation checks, then Bob's your Uncle.
There is an extended discussion of Barbarian Optimization at this guide here. That author rates the VHuman as the next best thing to half orc, alongside Dragon Born, Goliath, and Mountain Dwarf.
The Warforged Juggernaut from Wayfinder's Guide to Eberron (p. 68-70) gains +1 Constitution and +2 Strength (both stats are extremely useful to a barbarian). The unarmored defense is crazy (light armor gives you an AC of 11 + Dex mod + proficiency bonus, medium is 13 + Dex mod (max of 2) + proficiency bonus, heavy is 16 + proficiency bonus).
You have resistance to poison and advantage against being poisoned. You also cannot become diseased and you are technically always awake to protect your team as they sleep at night. You don't suffer from exhaustion from lack of rest (can be helpful to a berserker barbarian). And your carrying capacity is amazing! You count as a large creature for carrying capacity, which is relevant if you use carrying capacity rules. These all add up to make you into the group's powerhouse as you aren't affected by environmental dangers in those forms; you can carry the group's gear and make sure everyone is happy, and stay on watch at night when they feel sleepy.
And finally your unarmed damage is 1d4! (Not really that great, but it is very fun for the sake of roleplaying and bar fights.)
Plus, who wouldn't wanna be the Terminator in D&D?
Pros:
They have good unarmored defense (14+Constitution modifier if I'm not mistaken),
get points in Strength (razorback tortle)
get points in Constitution (desert tortle)
Can deal extra 1d4 of damage after every attack
These are good features to fit with the Barbarians
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