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Complete Monster: Baron Wolfgang von Strucker, Nick Fury's Arch-Enemy (as well as frequent rival of Captain America, Iron Man, and The Avengers), has the dubious honor of being both a Nazi war criminal and an international terrorist as the leader of HYDRA. A German noble who subscribed to racial superiority, Strucker achieved a high rank in the Death's Head and earned a personal enmity with Nick Fury himself. In a crowning moment of monstrosity on World War 2, Strucker found a race of benevolent, empathic aliens that crashed near the small German town of Gruenstadt. To cover his tracks, Strucker killed the entire population of the town and planned to manipulate the aliens into providing him their technology. As head of HYDRA, Strucker ordered countless terrorist attacks throughout the globe that claimed the lives of many, many innocent people, all the while secretly controlling S.H.I.E.L.D. through Life Model Decoys and playing his enemies to get many agents killed while HYDRA continued its plans unabated. Strucker's ultimate goals are global domination and genocide of all 'lower races', and he made many attempts at both. When informed his son Andreas had been murdered by Norman Osborn, Strucker's only response was that he owed Osborn a favor. He also killed his eldest son with the Death Spore Virus, which is just as awful as it sounds: it has bonded to Struckerís body and allows him to kill personally through his body. In addition, he was once The Dragon to Red Skull.
Fountain of Expies: Cobra from G.I. Joe is based on it. As a result, all Cobra Expies are Hydra Expies.
The Galactor organization from the anime Gatchaman comes close to having its Serial Numbers Filed Off. The Galactor troops wear green uniforms and red eyed cowls just like Hydra. The only major difference is that Galactor uniforms, in the spirit of The Seventies, have bellbottoms.
Hydra is an in-universe example of this with AIM, the Secret Empire, the Hand, and other groups being originally branches of the organization.
Moral Event Horizon: The fact they're lead by a succession of former Nazis should tell you something to begin with but if you had to pick, there's the fact they're responsible for the original Red Skull coming back.
Blowing up the SHIELD hellicarrier for the first time was another one, especially as it was filled with hundreds of junior agents.
Baron Strucker also rigged himself so that if he ever died, the "Death Seed" plague inside him would kill the rest of the world.
Marvel Cinematic Universe
Evil Is Cool: They're quite competent compared to other organizations and their comic book counterpart.