When HYDRA first appeared in the pages of Marvel Comics in 1965 (Strange Tales #135) it was as one of many Secret Society villain organizations (like the rest, little more than a copyright-skirting "borrowing" of Ian Fleming's 007-bedeviling SPECTER) set up to be crushed by Nick Fury and S.H.I.E.L.D. Led by nondescript evil businessman Arnold Brown, they were defeated in short order and clearly never intended to become a permanent or even important fixture of the broader Marvel Universe.
But as the revival of World War II-era Marvel heroes like Namor and Captain America inevitably brought the return of WWII-era villains like The Red Skull, Baron Von Srucker and Helmut Zemo, HYDRA's unforgettably-goofy iconography (that double-fist salute! Those H-strapped uniforms! That ridiculous skull-octopus insignia!) were soon found to have an invaluable use: A way for Marvel creators of the time (many of them Jewish, WWII veterans or both) to tell stories of the scariest thing they could imagine - the return of The Nazis - without plastering comics aimed primarily at children with swastikas and other explicitly fascist imagery.
A live-action incarnation of HYDRA is similarly baked into the continuum of the Marvel Cinematic Universe from the very (chronological) beginning, with a featured villain spot in Captain America: The First Avenger, and its surprise return in that film's sequel turned the "Hail HYDRA!" catchphrase into the 21st century's go-to reference point for secret, collusive corruption. From that point forward, the group's most regular presence has been on TV as the primary antagonists of Agents of S.H.I.E.L.D. Though officially thought eliminated after the events of the series' third season, season 4's final wave of episodes has inventively brought them back via the main characters finding themselves trapped in a computer-simulated alternate reality where HYDRA not only endured but now dominates the United States security apparatus.
With HYDRA back for at least one more (however simulated) hurrah on S.H.I.E.L.D. and taking center stage in an ultra-controversial story in the pages of Steve Rogers: Captain America, here's your guide to the history of The Marvel Cinematic Universe's most reliable nefarious collective:
The history of HYDRA in the MCU is inextricably bound to that of The Inhumans, a species genetically descended from early humans who were experimented on by The Kree in an attempt to create bio-weapons. Though born indistinguishable from "normal" humans, when exposed to the rare element Terrigen an Inhuman develops unique superhuman abilities (and, sometimes, a nonhuman appearance) through a process called Terrigenesis.
One of the most powerful early Inhumans, the parasitic telepath Hive, led a rebellion that displaced their Kree overlords from Earth, but Hive eventually grew so powerful himself that he amassed a legion of human followers who revered him as a living god. To prevent Hive (who had learned to increase his powers by devouring human beings for their life energy) from dominating the Earth himself, the other Inhumans used a Kree relic called The Monolith to create a makeshift black hole and banish him to the far-off planet Maveth.
However, the cult of Hive-worshipers endured after his exile; coalescing over centuries into a secret occult organization dedicated to masterminding their god's return to Earth and adopting his tentacle-draped face as a stylized demonic symbol. They come to call themselves HYDRA, in reference to a beast of Greek mythology said to sprout two new heads for each one that an attacker is able to remove.
ANCIENT HISTORY & EARLY MODERNITY
Though shrouded in secrecy, the HYDRA cult continued to evolve through the centuries - with leaders maintaining the singular focus on returning Hive to Earth, but also amassing power and influence by infiltrating and spreading through the political, business and familial aristocracies of each successive human civilization in the guise of more nebulously power-seeking goals.
Eventually, one of the especially powerful sects in 1800s Great Britain comes to possess The Monolith and begins a ritual practices of selecting believers via lottery to be sent through the black hole as emissaries (read: human sacrifices) to Hive. This sect eventually comes to be controlled by the Mallick Family, who consolidate power as one of the most prominent families within the HYDRA inner circle.
Over time, the true origins and belief-system of HYDRA's founders becomes less widely known among it's non-elite membership; until it is largely indistinguishable from (and, in certain cases, intertwined with) other secretive societies like Freemasonry or the Arena Club a.k.a. "The Council of Nine," - a group of wealthy businessmen who claim responsibility for the Black Tuesday stock market crash and the assassination of President McKinley, whose connection to HYDRA itself has been implied but not explicitly confirmed.
WORLD WAR II
As happens with most religions and/or cults, eventually HYDRA undergoes an ideological schism - which comes to a head in the mid-20th Century aid the outbreak of the Second World War.
While much of the elite HYDRA ruling class remains committed to being a secretive society working in the shadows to advance the ancient cause of restoring Hive's "godhood" on Earth, an increasing number of sects and individual power-brokers grow less committed to the supernatural "myths" of HYDRA's founding than to the earthly power they can accrue for themselves through political infiltration and a combination of ancient knowledge with mastery of advanced modern science. This more technocratic-inclined HYDRA abandons secrecy and reveals itself, aligning with Hitler's Germany in the guise of a Nazi "Deep Science" Division initially promising to provide the Third Reich with superweapons.
The "Deep Science" HYDRA is led by Johan Schmidt, a.k.a. "The Red Skull," who dedicates himself to finding and unlocking the secrets of The Tesseract - an ancient Asgardian treasure later confirmed to be a containment vessel for The Space Stone, one of the six all-powerful Infinity Stones. Though The Red Skull is defeated by the combined efforts of Captain America and the Strategic Scientific Reserve, he is far from the only seat of leadership in the Nazi incarnation of HYDRA.
One of Schmidt's lieutenants, Werner Reinhardt, is dispatched to Asia during the war to investigate discoveries of artifacts relating to ancient Kree visitors and the early Inhumans; in particular excavating a preserved Kree corpse and an Obelisk - one of multiple Kree vessels containing reserves of Terrigen in powerful crystalized form that is brings death to any human who touches it. A human-experimentation enthusiast in the vein of his Nazi contemporary Josef Mengele, Reinhardt conducts research trials on captive subjects attempting to work out the true purpose of the object, eventually discovering a young peasant girl named Jiaying who is able to touch The Obelisk with no ill-effects.
THE COLD WAR & S.H.I.E.L.D. INFILTRATION
Werner Reinhardt's experimenting is cut short when his compound is raided by SSR Agent Peggy Carter and the Howling Commandos. Reinhardt is imprisoned for life, visited on occasion only for interrogation, while his research materials are taken into custody by the SSR, which also comes into possession of the The Tesseract when it is rediscovered by Howard Stark during his unsuccessful searches for the body of the presumed-dead Captain America. It is believed that this brings an end to HYDRA, as modern history has only ever known of the organization in its Red Skull-led Nazi version.
One member of HYDRA manages to escape both death and imprisonment: Arnim Zola, The Red Skull's weapons-designer confidant, is among the ex-Nazi scientists brought to work for U.S. interests under the (real-life!) Operation Paperclip program. Supposedly renouncing his ties to HYDRA, Zola (who, while under detention by the SSR, has also brought the dangerous ex-Soviet psychiatrist and hypnotist Johann Fenhoff, a.k.a. "Doctor Faustus," into the fold) becomes a major figure in the SSR's Cold War transformation into S.H.I.E.L.D. This turns out to have been a bad idea.
Under supervision of Arnim Zola, HYDRA loyalists revive and transform Captain America's thought-dead ally James Buchanan "Bucky" Barnes into a cyborg assassin codenamed The Winter Soldier. While under their control, he is dispatched stage the assassinations of Howard and Maria Stark in order to steal crucial research information - leaving their son Tony an orphan and heir to the family fortune.
His body ravaged by cancer, Zola's brainwaves are reconstituted as an artificial-intelligence system that becomes the defacto secret leader of a revived version of The Red Skull's HYDRA operating, recruiting and quietly growing within the ranks of S.H.I.E.L.D. itself. The "elite" powerbrokers of the original HYDRA, however, are also still around and aware of these changes; manipulating S.H.I.E.L.D. in their own way over the decades and culminating in the ascension of Gideon Mallick (who meets with the imprisoned Werner Reinhardt but declines to re-establish equal ties between the two HYDRAs) to head of the World Security Council - which overseas much of S.H.I.E.L.D's global operations.
Werner Reinhardt is freed from prison in 1989 by HYDRA-loyal S.H.I.E.L.D. Agents and resumes his original research, eventually rediscovering Jiaying and confirming his original suspicions: She is an Inhuman with healing powers, and has not physically aged since they last met. Reinhardt vivisects her alive, eventually extracting her organs and DNA and implanting them into himself - which results in his boidy de-aging into a middle aged form. He re-names himself "Daniel Whitehall" and becomes the (non-computerized) human leader of HYDRA-within-S.H.I.E.L.D.
Unknown to Reinhardt/Whitehall, Jaiying's body is repaired and revived, thanks in part to her Inhuman powers, by her husband Dr. Calvin Johnson (with whom she has had a daughter, future S.H.I.E.L.D. Agent Daisy "Skye" Johnson) though the experience has left her in a psychotic state seeking the extermination of the human race. Calvin, looking to aid her, ultimately transforms himself into the supervillain Mister Hyde and make secret underworld alliances with HYDRA in order to get close to Whitehall and exact revenge.
In the mid-1990s, HYDRA operative John Garrett recruits Grant Ward into Hydra-within-S.H.I.E.L.D. and masterminds his placement onto an elite team led by secretly-ressurected Battle of New York veteran Agent Phil Coulson. Garrett, posing as a made-up supervillain "The Clairvoyant," also sets up the HYDRA front C.E.N.T.I.P.E.D.E. ostensibly to create super-soldiers but in reality to help a badly battle-damaged Garrett suss out the mysterious serum (derived from the Kree corpse taken from Reinhardt by Agent Carter in WWII) used to revive Coulson.
Alexander Pierce, former U.S. Secretary of Defense and chair of the World Security Council, reveals himself as a HYDRA sleeper who attempts to carry out Arnim Zola's dream of using S.H.I.E.L.D. infrastructure to conquer the planet. After attempting the assassination of Nick Fury by The Winter Soldier, Pierce, Garrett, Ward and other secretly HYDRA-aligned Agents reveal themselves in an attempted coup of the U.S. government that is thwarted by Captain America, The Black Widow and The Falcon in Washington D.C. and the efforts of Phil Coulson's S.H.E.I.L.D. splinter-team - which becomes the new "covert" S.H.I.E.L.D. when the official organization is subsequently dismantled.
HYDRA itself, however, survives the coup; with Gideon Mallick reasserting power among "original HYDRA's" elite and Baron Wolfgang Von-Strucker conducting research on alien artifacts stolen from within S.H.I.E.L.D. itself (including Loki's scepter, which contains the "Mind" Infinity Stone) in the remote nation of Sokovia. Though his operation is terminated by The Avengers, his work leads to the transformation of two Sokovian orphans into The Scarlet Witch and Quicksilver.
THE RETURN OF HIVE
Phil Coulson's "Secret S.H.I.E.L.D." comes into conflict with Daniel Whitehall over possession of the last Kree Obelisk, which leads to Whitehall's death at Coulson's hands, Agent Daisy Johnson's discovery of her true origins, her transformation into the Inhuman superheroine "Quake" and her reconnection with her mother Jaiying in the Inhuman sanctuary of Afterlife. When Jiaying betrays her family and attempts to start a war with S.H.I.E.L.D., it leads to the discovery that the last remaining S.H.I.E.L.D. battleship had been serving as a floating safehouse for The Monolith; which is later accidentally activated and pulls Agent Jemma Simmons through the portal to Maveth. She survives there for several months with the help of an astronaut who was marooned there by a HYDRA-backed mission.
Grant Ward, having escaped S.H.I.E.L.D. custody and begun a one-man war on the organization, goes looking to revive HYDRA on his own terms to mixed success; while Phil Coulson sets in motion a series of plans that systematically eliminate the known remainders of HYDRA leadership, while other former HYDRA operatives connected to The Winter Soldier Program begin to be murdered by a mystery assassin who turns out to be an aggrieved Sokovian military veteran named Helmut Zemo. During this period, HYDRA also backs the efforts of Darren Cross to mass-market Hank Pym's shrinking technology, which is thwarted by Scott Lang aka the new "Ant-Man."
Jemma Simmons is rescued from Maveth and enlists her S.H.I.E.L.D. allies to re-open the portal and attempt a rescue of her astronaut friend Will Daniels; who unknown to her has been killed and become the new physical host of Hive. Amid this, Gideon Mallick seeks out Grant Ward and brings him into the fold of the "real" HYDRA and reveals the truth about their Hive-worship origins to him - promising him a powerful place within the organization if he can help hijack S.H.I.E.L.D's use of The Monolith to complete their ancient mission. Ward ultimately travels to Maveth, but so does Coulson - who murders Ward in a moment of weakness that ultimately allows Hive to claim Ward's body as a host and escape to Earth.
Hive, the body of Grant Ward, aligns with Gideon Mallick and takes control of the remainder of HYDRA. He declares war on S.H.I.E.L.D., attempting to draw the increasing population of newly-emerging Inhumans (transformed by a supply of Terrigen being released into the world's water supply by Jiaying during her failed attempt to forment war between Inhumans and S.H.I.E.L.D.) to his side and use a Terrigen Bomb to transform all humans and Inhumans alike into a compliant "subspecies" loyal only to him.
THE DEATH OF HIVE - THE END?
Though his mind-control powers briefly allow Hive to take control of Agent Daisy Johnson (whom he manipulated into killing a traitorous Gideon Mallick), he is eventually killed by the electricity-powered Inhuman Lincoln, who sacrifices himself to destroy the monster - seemingly eliminating HYDRA for good.
The death of Hive, coming after the would-be Inhuman god had already eliminating much of the remaining HYDRA himself, would seem to have been the end for the organization - and indeed, HYDRA has not shown its face again (in the real world) for the duration of the series fourth season. But just like in the comics, it feels as though a return for HYDRA is never fully out of the question.
Many Marvel fans theorize (and/or hope) that The Red Skull was merely displaced from time rather than being destroyed by The Tesseract at the end of Captain America: The First Avenger, and if he was to return it feels rather likely he'd attempt to bring "his" HYDRA back into existence - especially as it's never been explicitly declared whether Johann Schmidt knew about or "believed in" Hive in the first place. Perhaps more likely, the HYDRA of "The Framework" (the virtual-reality simulation the Agents of S.H.I.E.L.D. currently find themselves trapped in) is led by an avatar of the simulation's LMD android creator Aida; who has proclaimed herself "Madame HYDRA." Given that Aida has already transferred her own consciousness into other LMD bodies before, it's not unreasonable that - whatever happens in The Framework - she might decide to try making that role work out in the real world, too.
Plus, HYDRA is such a malleable entity as a foil, it's not difficult to imagine future Marvel filmmakers wanting to bring back at least a branch or a handful of them to serve as bad-guys for those situations where an Evil Organization is just helpful to have around. Bottom line: much like in the original comics, it's pretty unlikely we've heard the last of HYDRA.