A half-orc is exactly what it says on the tin - a fusion of human and orc. Traditionally, such procreation is done under a male orc/female human paradigm, predominantly under connotations of rape (not explicitly stated as such, but heavily implied nonetheless). This has made the half-orc one of the more traditionally "edgy" and politically incorrect racial options, so it has almost as many haters as it did fans; the fact that half-orcs are usually described as looking more like orcs (aka, monstrous) has further made them a somewhat marginal race even amongst edgelord players, who are more likely to gravitate towards races like tieflings, dhampyres, or even half-elves (who were actually also usually the result of rape in D&D 1e). In fact, this "child by rape" standard origin is generally held up as the reason why it's believed TSR didn't make half-orcs playable in Advanced Dungeons & Dragons until the Complete Book of Humanoids, when they had first appeared in the 1e PHB.
Traditionally, half-orcs receive a "watered down" version of the orc's traits; this amounts to a lesser Strength boost but also a lessened penalty to mental stats, as well as by having better nightvision than humans, but also better daylight vision than orcs, gaining darkvision without suffering light sensitivity (except in 3e, where they still had the orcy extra vulnerability to spells like Sunlight). Bonuses towards intimidation-type effects, reflecting their frightening mien, aren't uncommon, and they tend to lean towards evil and chaos in settings where orcs just swing that way for reasons of giving PCs critters to kill without feeling bad about it.
Weirdly, half-orcs weren't quite in line with the aforementioned paradigm in AD&D 2e; they got +1 to both Strength and Constitution to offset their -2 Charisma, with pureblood-orcs only getting the +1 Strength, and they had higher racial maximums for Constitution and Intelligence. Meanwhile, orcs had a higher racial maximum for Wisdom and could get to much higher levels in Cleric, Shaman, Witchdoctor and Thief than half-orcs (unless the half-orcs had exceptional ability scores). Pureblood orcs had a chance to notice new/unusual construction and sloping passages, whilst half-orcs lacked the light aversion of the purebloods. In the 1st edition Player's Handbook, half-orcs were allowed unlimited level advancement in the assassin class, as well as the ability to multi-class as assassin/fighters, or assassin/clerics.
The half-orcs suffered considerably when converted to 3e, which is responsible for cementing the archetype of the half-orc as a "big dumb brute" in most /tg/ circles. With +2 to Strength contrasted by -2s to Intelligence and Charisma, and their only unique racial traits being darkvision, "orc blood" (can use orc-exclusive feats and magic items, treated as an orc for racial triggered abilities & spells) and a favored class of barbarian, they were the most mediocre and underpowered race in the PHB. They didn't even make very good assassins any more! Having no bonus to Dexterity (the core class-based ability score) was one thing, but that Intelligence penalty severely hampered their ability to get the skill ranks they needed to even qualify for the class. As a result, even more so than small races like halflings and gnomes, half-orcs tended to be unpopular and pigeonholed.
It got so bad that both Pathfinder and 4th edition tried to fix it, with 4e even leaving half-orcs out of the PHB in order to devote more time to coming up with a good niche for them - something that wasn't popular with many people, but really wasn't unprecedented.
The PF half-orc is basically just the 3.5 version with some extra orcy racial traits; swapping the +2 Str/-2 Int/-2 Cha paradigm for "+2 to one ability score of the player's choice", and gaining +2 to Intimidation checks, Weapon Familiarity (Double-Axe, Falchion, Ork), and the Orc Ferocity racial trait (can fight on for 1 round after dropping to 0 hitpoints). It was a simple fix, but compared to the 3.5 half-orc, it was a huge step up. Plus, if nothing else, they have access to the awesome scarred witchdoctor archetype for witches, which is both crunchy goodness and awesomely flavored.
The 4e half-orc, in comparison, became a +2 Str/Dex race with bonuses in Endurance and Intimidate, Low-Light Vision, the Half-Orc Resilience trait (gain +5/10/15 temporary hit points the first time you are bloodied in an encounter), which makes them a lot tankier, the Swift Charge trait (+2 speed when charging), which lets them excel at running people down, and the Furious Assault racial power (1/encounter, boost weapon damage dice for a successful hit by +1 dice). All in all, they're melee monsters, made for kicking ass and taking names, but they don't have to be barbarians; 4e half-orcs make pretty good rogues, thus restoring their 1e traditional expertise, and surprisingly good monks.
5th edition followed in 4e's footsteps and focused on making them natural ass-kicking tanks, almost like being a barbarian-lite just by your choice in race. +2 Str and +1 Con, Darkvision, free Proficiency in Intimidation, can tank a killing blow 1/day, and +1 dice worth of damage on critical hits with melee weapons. Maybe not as broad in possibilities as the 4e version, but certainly a hell of a lot stronger than the 3e version.
Of course, bad crunch was only one half of the half-orc's problem. As we said up at the top, they've always been the edgiest of the "traditional corebook" races. Whilst PF ran with it, even emphasizing the whole "you were probably born of rape and everybody hates you!" stuff, other editions or even settings within 2nd and 3rd edition have tried to move away from that rubbish.
For example, there was implying that female orcs generally aren't as ugly as people make out, and that frontier clans tend to intermarry with some frequency. Even way back in AD&D, there were occasional comments implying individuals having peaceful co-relations; the factol of the Bleakers in Planescape was the son of a blind man and his orc wife, whilst Greenwood has spoken on forums about isolated regions where human widows or spinsters have accepted displaced orc braves as, essentially, live-in hired help that accepts sex and food in lieu of money for taking care of the homestead. Some places actually have full on half-orc populations, where man and orc have so thoroughly interbred that everyone's at least a little bit half-orc. Fourth edition also introduced more explicit ideas for racial backstories beyond "orcs like to rape and pillage", such as half-orcs being the result of deliberate intermarriage on a massive scale, crossbreeding by some external faction, or even a deliberate creation of either Gruumsh the orc-god to create a superior strain of orcs to lead the rest of his children to victory, or Kord to create a super-race of warriors.
Everyone agrees that this is in much better taste and shuts out some of the edgelord bullshit, and it's gone over quite well.
Of course, it probably goes without saying that exactly how oppressed and angsty half-orcs are varies with the setting, and, more to the point, with the setting's orcs.
When they're just traditional rampaging barbarians all the time, they tend to get all kinds of shit on -- Golarion, home of the Pathfinder setting, really plays up the "half-orcs are usually born to rape" in the fluff, so they get a lot of flak... somewhat two-facedly, though, most of their important half-orc characters are not rape-children, and in places like the deserts or the Mwangi jungle, half-orcs are actually quite respected. Desert half-orcs actually get bonuses to diplomacy instead of intimidation.
In places like Eberron, where orcs aren't so bad once you get to know them, they fare much better, though still suffer discrimination due to a perceived lack of intelligence.
In some places, it might even vary from place to place. Forgotten Realms, for example, has the Eastern territory of Thesk; when the Tuigan Horde nearly overran the place, Thay sent an army of orcs to defend it, and then abandoned them when the Horde was pushed back. The Theskians took them in and the orcs actually settled down pretty well - in Thesk, the stereotype of the half-orc is a gruff, hard-working rancher or miner, and a damn good neighbor to have in a place where bandits and invasions happen all the time. Meanwhile, in the Northern territories of Faerun, half-orcs are rare and usually killed at birth, because they're seen as being no different than the local orcs, who are murderous, wrathful monsters - although this has softened over editions, thanks to Many-Arrows.
Almost every setting, though, points out that anyone big and beefy enough can carve out a niche for him- or herself in among the "civilized" races, and that being the smartest motherfucker in the room and only slightly less strong has its own advantages in the orc tribes. Lots of famous orcish heroes had enough human blood in them to make them, as their spacefaring cousins would say, "ded kunnin'."
Although this page focuses on human/orc hybrids, in AD&D, Half-orcs were a lot more diverse. The "basic" half-orc entry in the Monster Manual covered not only orc/human hybrids, as discussed in detail here, but also orc/goblin and orc/hobgoblin crossbreeds. Orc/ogre crossbreeding was rumored to be the source of the Orog species (although 3.5 Forgotten Realms retconned orogs as an Underdark dwelling species of bigger, smarter orcs), which itself received magically augmented/created variants in the "Neo-Orogs" of the Forgotten Realms (divided between Red ones, for fighting, and Black ones, for assassination). A confirmed orc/ogre crossbreed, though definitely leaning towards the Ogre (it was actually listed under "half-ogre" in the AD&D MM) is the Ogrillon, which basically resembles a giant orc covered in bony spikes. The weirdest half-orc is the Losel, or "ape-orc" of Greyhawk, which is half-orc and half baboon -- thankfully, that one's believed to be a magical creation, like the owlbear.
Heck, in 1st edition, there were actually some stats given to the orc-bugbear, the orc-gnoll, the orc-goblin, the orc-hobgoblin, the orc-kobold, and the orc-ogre crossbreeds, although these were presented more as new enemy fodder, in the 44th issue of Dragon Magazine.
That said, the 5th edition monster manual's lore for orcs does say that orcs can make half-orcs with many different races of compatible size, explicitly calling out dwarfs as a candidate, so with a DM's permission, a half-orc's fluff could get... creative.
Non-human half-orcs with actual mechanical support have appeared here and there. The Tel-Amhothlan from Kingdoms of Kalamar is a half-elf half-orc. The Dworg is a half-dwarf half-orc]] from the Midnight campaign setting.
- Krusk, the iconic half-orc from Dungeons & Dragons 3rd Edition. A pretty swell guy.
|Dungeons & Dragons 2nd Edition Races|
|Core:||Dwarf - Elf - Gnome - Half-Elf - Half-Orc - Halfling - Human|
|Dark Sun:||Aarakocra - Half-Giant - Mul - Pterran - Thri-kreen|
|Dragonlance:||Draconian - Irda - Kender - Minotaur|
|Mystara:|| Aranea - Ee'ar - Enduk - Lizardfolk (Cayma - Gurrash - Shazak) |
Lupin - Manscorpion - Phanaton - Rakasta - Tortle - Wallara
|Oriental Adventures:||Korobokuru - Hengeyokai - Spirit Folk|
|Planescape:||Aasimar - Bariaur - Genasi - Githyanki - Githzerai - Modron - Tiefling|
|Spelljammer:||Dracon - Giff - Grommam - Hadozee - Hurwaeti - Rastipede - Scro - Xixchil|
|Ravenloft:||Broken One - Flesh Golem - Half-Vistani - Therianthrope|
Book of X:
| Alaghi - Beastman - Bugbear - Bullywug - Centaur - Duergar |
Fremlin - Firbolg - Flind - Gnoll - Goblin - Half-Ogre - Hobgoblin
Kobold - Mongrelfolk - Ogre - Ogre Mage - Orc - Pixie
Satyr - Saurial - Svirfneblin - Swanmay - Voadkyn - Wemic
|Dragon Magazine:||Half-Dryad - Half-Satyr - Uldra - Xvart|
|Dungeons & Dragons 4th Edition Races|
|Player's Handbook 1:|| Dragonborn - Dwarf - Eladrin - Elf |
Half-Elf - Halfling - Human - Tiefling
|Player's Handbook 2:||Deva - Gnome - Goliath - Half-Orc - Shifter|
|Player's Handbook 3:||Githzerai - Minotaur - Shardmind - Wilden|
|Monster Manual 1:|| Bugbear - Doppelganger - Githyanki |
Goblin - Hobgoblin - Kobold - Orc
|Monster Manual 2:||Bullywug - Duergar - Kenku|
|Dragon Magazine:||Gnoll - Shadar-kai|
|Heroes of Shadow:||Revenant - Shade - Vryloka|
|Heroes of the Feywild||Hamadryad - Pixie - Satyr|
|Eberron's Player's Guide:||Changeling - Kalashtar - Warforged|
|The Manual of the Planes:||Bladeling|
|Dark Sun Campaign Setting:||Mul - Thri-kreen|
|Forgotten Realms Player's Guide:||Drow - Genasi|
|The Races of Pathfinder|
|Player's Handbook:||Dwarf - Elf - Gnome - Half-Elf - Half-Orc - Halfling - Human|
| Aasimar - Catfolk - Changeling - Dhampir - Duergar |
Drow - Fetchling - Gillman - Goblin - Grippli - Hobgoblin
Ifrit - Kitsune - Kobold - Merfolk - Nagaji - Orc - Oread
Ratfolk - Samsaran - Strix - Suli - Svirfneblin - Sylph
Tengu - Tiefling - Undine - Vanara - Vishkanya - Wayang
|Bestiaries:|| Android - Astomoi - Caligni - Deep One Hybrid - Gathlain |
Gnoll - Kasatha - Munavri - Naiad - Orang-Pendak
Reptoid - Rougarou - Shabti - Trox - Yaddithian
|Adventure Paths:||Being of Ib - Kuru|
|Inner Sea Races:|| Ghoran - Monkey Goblin - Lashunta - Skinwalker |
Syrinx - Triaxian - Wyrwood - Wyvaran
|Ultimate Wilderness:||Vine Leshy|
|Blood of the Sea:||Adaro - Cecaelia - Grindylow - Locathah - Sahuagin - Triton|
|Planar Adventures:||Aphorite - Duskwalker - Ganzi|
|The Races of Starfinder|
|Core Races:||Android - Human - Kasatha - Lashunta - Shirren - Vesk - Ysoki|
|Legacy Races:||Dwarf - Elf - Gnome - Half-Elf - Half-Orc - Halfling|
|Expanded Races:|| Aasimar - Astrazoan - Bantrid - Barathu - Bolida - Borai - Brakim|
Contemplative - Copaxi - Damai - Draelik - Dragonkin - Drow - Elebrian
Embri - Endiffian - Ferran - Formian - Ghibrani - Ghoran - Goblin
Gosclaw - Gray - Haan - Hobgoblin - Ifrit - Ikeshti - Ilthisarian - Kalo
Kanabo - Kayal - Khizar - Kish - Maraquoi - Morlamaw - Neskinti - Nuar
Orc - Oread - Osharu - Pahtra - Phentomite - Quorlu - Reptoid - Ryphorian
Sarcesian - Screedreep - Scyphozoan - Selamid - Seprevoi - Shimreen
Shobhad - Skittermander - SRO - Stellifera - Strix - Suli - Svartalfar - Sylph
Tiefling - Trox - Undine - Uplifted bear - Urog - Verthani - Vilderaro - Vlaka
Witchwyrd - Woioko - Wrikreechee