It doesn't stand for anything, it's not an acronym, but they're a self-important organization so they like to have it in all caps where possible. Von Strucker took the name from the many-headed beastie Hercules fought back before there were super-heroes populating every corner of the planet.
Instructed by his boss, the Red Skull, to form a fascist power base in the Far East, Baron Von Strucker developed a partnership with a Japanese subversive group which became the nucleus of the original HYDRA. After World War II, the many-tendriled organization became a powerful and dangerous force against the free world as it actively worked to over-throw governments and undermine economies. HYDRA collapsed after the destruction of Hydra Island by Nick Fury; however, splinters of the group remained and eventually HYDRA was reformed. During the early stages of this reconstruction it seemed as if HYDRA had abandoned its fascist policies and would achieve world dominance more through illicit commerce and criminal terrorism than political and ideological terrorism. The resurrection of Baron Von Strucker also meant a rebirth of the Nazi rhetoric the group was founded upon.
HYDRA's many members are required to be athletic and well-versed in hand-to-hand combat, firearms, and advanced technology. In the early days, particularly brilliant members were shunted off into A.I.M., the equivalent of a terrorist think-tank. Much of HYDRA's futuristic weaponry is based on extra-terrestrial Gnobian technology, which Von Strucker came into contact with in the waning days of World War II.
HYDRA is the opponent of all defenders of freedom, which of course means Captain America, the Avengers, Nick Fury and S.H.I.E.L.D. just can't stay away from anyone wearing that tacky green uniform. In recent years, HYDRA has also had to deal with such upstarts as Silver Sable and her Wild Pack, Iron Fist, DareDevil and even former member the Kingpin.
A number of distinctive criminals have been associated with HYDRA over the years. Back before he started looking at the world through Rose-colored glasses, Richard Fisk headed up their Las Vegas operations. Another Spidey mainstay, Silvermane, became the Supreme HYDRA when he got fed up with Maggia infighting, but scurried back to his mob job when he realized there were worse things out there than Spider-Man. Viper once called herself Madame Hydra, and still tints her hair green as a salute to the old HYDRA color scheme. The Grim Reaper may well have gotten his start with the organization, and once co-led the New York branch of HYDRA. Jessica Drew, the original Spider-Woman, trained with the group but blew her first assignment when she couldn't bring herself to kill Nick Fury. HYDRA also uses free-lance super-powered operatives when it needs to, such as the Chameleon, the Space Phantom, and Skeleton Ki.
Far from being a covert terrorist group, HYDRA thrives on grandstand plays. They have twice managed to hold the planet hostage, once with the Betatron Bomb, and later with the Death Spore Bomb. They also achieved a coup of astonishing brilliance and brutality when they murdered the entire graduating class of S.H.I.E.L.D., eliminating 1500 new agents before they had a chance to became threats.