"Blood for the Blood God! Blood for the Blood God! BLOOD FOR THE BLOODGAWDRARAWRERAR!"
Khorne Berzerkers (languagefaggotry note: that should be Khornate Berzerkers, but "Khorne Berzerkers" is the official title) are Space Marines dedicated to Khorne, the god of blood, war, and fucking your shit up.
Most Khorne Berzerkers come from the World Eaters Traitor Legion. Their Primarch, Angron, decided that his Marines should have partial lobotomies, just like himself. When this was done, the only emotion they could feel was RRRAAAAGGGGEEE. As the Horus Heresy dragged on, the World Eaters would eventually turn to the worship of Khorne, with the Berzerkers leading the way. Of course, they lost, fled to the Eye of Terror, and have spent the past 10,000 years worshiping Khorne the only way they know how: BATTLE, GLORIOUS BATTLE! Some renegades have joined them, but very few Space Marines are angry enough to keep up. They usually hang out in small squads to company to chapter size groups although the bigger formations are always temporary; there is too much RAGE going on in their minds to work together for long, because, after all, they are as likely to spill their comrades blood as an enemy's. A point of contention among them is whether or not a Khornate Berserker is still serving the Blood God if he spills blood using ranged weapons.
Berserkers (or Berzerkers if you prefer) trace their origins back to Rogue Trader and are arguably one of the few units from then to basically survive the retconning rampant throughout other units in the game, largely in terms of fluff and spirit. Their crunch and effectiveness however have varied throughout the editions but they have always been described as angry motherfuckers that have no fear, just all anger - built for close combat exclusively.
This is the one neckbeards think happy thoughts about. Frenzied, shitload of attacks, chainaxes, kickass weapon skill - mmmmm... Notable here was that these basically ported rules over from Fantasy and the Aspiring Champion could take a Bloodletter's sword for gear. Fun times!
This is where things get confusing. Though 3E nerfed everything when it landed and the Berserkers were no exception, there was a stitch in the rules - just how much strength does an individual Berserker have? Some said 5, some said 6: again, the rules said they had the Mark of Khorne which granted +1S back in the day, but Games Workshop had to point out that this bonus was included in the profile. Regardless, they were stronger and had more attacks than a CSM as well as being fearless. Unfortunately, their signature chainaxes were effectively sticks so bummer about that. A quirk with this edition is that if your chaos lord had the Mark of Khorne, you could take these guys in the new troops slot, so that was cool too.
Then the infamous 3.5 book hit and... well, Berserkers actually didn't get blasted into the sky with cheese. They were good sure, but not seriously OP as they were just upgrades for various CSM units. Their chainaxes were Ork Choppas and the Mark of Khorne bestowed fearlessness, +1 attack and if played in the World Eaters army, lost all options for heavy and special weapons except plasma pistols. Basically, the Berserkers themselves weren't that great but at this time there were also Berserker bikers, chosen, terminators and possessed, which was pretty fucking cool. Oh, and some of those could take veteran skills, which was almost always Furious Charge because of course it was.
Aw shucks, well that happened didn't it? No more Berserker anythings but MEQs on foot with no extra sauce and their chainaxes are sticks again. However, they got moved to the Troops slot permanently which was cool plus they got a higher WS 5 and Furious Charge. Overall, not bad for a price bump to 21 pts./model but with all the things included, it was actually a bit of a steal from the previous codex. Unfortunately, they were overshadowed by Plague Marines, who were just more survivable for only costing 2 pts./ model more. Berserkers would languish for an edition before getting an update in 6th...
This is the painful part; they were basically regular Chaos Space Marines with close combat weapons and bolt pistols (which they use as good clubbing instruments). They retained their higher Weapon Skill, Fearless, and Furious Charge from the last time around, as well as the Mark of Khorne, which now gave them Rage and Counter-Attack. They could hypothetically butcher in melee if they could get there and assuming they had the charge. Their price tag got cut to 19 points per model, but with poor transport options (expensive Land Raider or assault-ramp-less Rhino), and then came the overall nerfing of melee in 7th Edition (and the sheer fragility of one wound MEQs in the current meta), it's very difficult to get them there. In their defense, the most Zerky CSM squad will be 20p cheaper, but not Fearless or at WS5. So it's always more worth it to take regular Zerks, which scored in 7th and changed Counter-Attack so it no longer requires a Ld test. Also MoK Lord or this fun guy made them Troops.
Berzerkers have been buffed back into the little close combat monsters they should always have been. They now have a base Strength of 5, which can be increased to 6 with a Chainaxe (which also has -1 AP). Alternatively a regular Chainsword (with no armor modifier) gives them an additional attack (on their 2 base). Chainaxes can either replace their Chainsword or their Bolt Pistol. Even better, the Blood For The Blood God ability allows Berzerkers to fight twice in each Fight Phase (yours and your opponent's), meaning a single Berzerker will be hitting 4 times at S6 or 6 times at S5 every fight phase depending on your loadout.
In Dawn of War series
Khorne Berzerkers came to Dawn of War series with Winter Assault. Because Chaos didn't have enough melee so far. They are performing pretty amazing in melee combat. Heavy infantry armor, no morale, satisfying melee damage. Pretty scary in melee combat for other melee units. Pretty similar to table top. However, their bleed is a problem if you keep reinforcing them. They also prefer that you drive faster or else this happens 
In Dawn of War 2, there isn't a unit called Khorne Berzerkers. Instead, you are upgrading your Chaos Space Marines squad with Mark of Khorne, equipping them with chainaxes and plasma pistols. Their Aspiring Champion gets the same axe, but with a melta pistol. Their axes are all doing power melee damage, deadly against all heavy infantry armours. They are also gaining increased speed and health. However, it is not a great thing to use them like an elite melee troop. They are effective if you use them like shock troops, like an Assault Marine Squad. Should be send against defenseless ranged squads to tie them up in melee combat. They are not winning against fully upgraded Sluggaz or Banshees in game sadly.
Warhammer Fantasy Age of Sigmar:
Surprisingly, Khorne has never had dedicated worshipper-troops in Warhammer Fantasy, just Chaos Warriors/Knights sporting his Mark, though they certainly fulfill the role fluff-wise. To be fair, you could always take Juggerknights, because who doesn't like a cavalry of bloodthirsty iron rhinos? The End Times adds two units even closer to the spirit, via the Archaon book (leaked early in White Dwarf). Both have the basic fluff of being particularly advanced and crazed Khorne worshipers who have started to mutate into even more deadly fighters, complete with growing muscles so big they're starting to burst out of their Chaos Armor.
They're both Infantry sporting a Movement 4, Weapon Skill 6, BS 3, Strength 4 (Strength 5, for Wrathmongers), Toughness 4, Wounds 3, Initiative 5, Attacks 3, Leadership 8 profile. The differences are subtle.
Skullreapers are a 40pts per model Special choice who run around sporting paired weapons (+1 Attack), which they can upgrade to Paired Ensorcelled Weapons (+1 Strength, attacks count as magic, +1 Attack) for +5 points per model. Since the minimum size of a squad is five, even without taking a Champion (who gets another +1 Attack), a bare-bones Skullreaper squad with Ensorcelled Weapons is costing you 225 points for something that can dish out 20 S5 hits that will actually hit damn Ethereal troops at Weapon Skill 6. Not too bad at chewing up basic mooks, all things considering.
Wrathmongers are even madder than Skullreapers; they cost 55 points each and use up a Rare slot, thanks to their sporting Paired Wrath-flails (+1 Strength in first round of combat - yes, that means they're S6 during the first round, extra attack, Impact Hits D3). Again, bare minimum squad is 5 for 275 points, which is dishing out 5D3 S5 Impact hits on the charge and then 15 S6 hits directly afterwards... pretty damn choppy.
The later Age of Sigmar setting created an entirely new subset of Khornate-exclusive warbands known as the Bloodbound, who are essentially Khornate Warriors of Chaos in more thematically unified armour, with their beard fetishes enhanced. There are the Blood Warriors, who feel such unmatched fury that it becomes a physically tangible force that emanates from their bodies like a heatwave. They are apparently so manly that they can continue to hack their enemies into tiny bits even while they're in their death throes. Slaughterpriests, who are basically Khornate Evangelical priests. Bloodreavers, basically your bald, psycho meatshields (read: Chaos Maraduers) with forked beards that should remind you of Flesh Hounds of Khorne. Bloodstokers, fat lion-tamers who drive their fellow warriors to untold heights of rage by being insufferable pillocks. Bloodsecrators, portal carriers (read: your bread and butter).
Non Warhammer Berserkers
The term "Berserker" comes from the Viking "Baer-sark", or "bear-shirt", referring to dreaded warriors who would dress themselves up in bearskins and likely worked as champions for a Lords or Jarls, fighting duels in their stead (Point of contestation, Baer is not perfectly translated and could mean bare as in no clothes at all). Some even believed they could transform into bears, similar to the Ulfsark or "Wolf-Shirt" were believed to be able to turn into wolves, making them a possible origin for the myth of the werewolf. Similarly frenzied warriors appeared in many European Bronze and Iron Age cultures, like the Celts and Picts, they however were not called berserkers, and even among vikings, people who went "berserk" weren't called berserkers in any of the sagas remaining to this day. However a 13th century Norse translation of a French story: "Yvain, Le Chevalier au Lion", translates the french word for champion into the Norse word for berserker.
The Barbarian roleplaying game class, particularly in Dungeons & Dragons and Pathfinder is based significantly on the concept of the common misconception of a Berserker, as well as on Conan (who himself by D&D lore is more of a Fighter with thief skills). Both games also have a Berserker-themed archetype for fighters; the Berserker in Pathfinder, and the Battlerager in 4th edition.
4th edition also has the Berserker as a Variant Class for the Barbarian, trading the Barbarian's features for a completely new set of features. Weirdly, it's not technically an "Essentials Class", as it still uses the AEDU System!
Berserker is also the name of a ridiculous song performed by Love Among Freaks that appeared in the movie Clerks in 1994.